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Mathematics · Class 12 Science

Ch 7Integrals — Class 12 Mathematics, concept-first.

Differential calculus centres on the derivative, originally motivated by the problem of defining tangent lines to graphs and calculating their slope. Integral calculus, by contrast, is motivated by the problem of defining and calculating the area of the region bounded by the graph of a function.

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Sine Double Angle Integration

Suppose you want the area under from to . This graph oscillates twice as fast as a regular sine wave, completing a cycle in units instead of — the "double angle" inside compresses the wave horizontally.

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7.1

Introduction

Differential calculus centres on the derivative, originally motivated by the problem of defining tangent lines to graphs and calculating their slope.

7.2

Integration as an Inverse Process of Differentiation

Differentiation gives us the rate at which a function changes. Integration reverses this: we start with the derivative and ask, "What original function could have produced this?" This reverse process…

7.2.1

Some Properties of Indefinite Integral

26 Q

This subsection establishes the fundamental properties of indefinite integrals — the rules you will use every time you break a complicated integral into simpler pieces.

+Worked Examplesi4 questions
  1. Example 1Write an anti derivative for each of the following functions using the method of inspection: (i) $\cos 2x$ (ii) $3x^2 + 4x^3$ (iii) $\dfrac{…Free
  2. Example 2Find the following integrals: (i) $\int \dfrac{x^3 - 1}{x^2}\, dx$ (ii) $\int \left(x^{2/3} + 1\right) dx$ (iii) $\int \left(x^{3/2} + 2e^x…Free
  3. Example 3Find the following integrals: (i) $\int (\sin x + \cos x)\, dx$ (ii) $\int \operatorname{cosec} x\,(\operatorname{cosec} x + \cot x)\, dx$ (…Preview
  4. Example 4Find the anti derivative $F$ of $f$ defined by $f(x) = 4x^3 - 6$, where $F(0) = 3$.Preview
+Exercise 7.1i22 questions
  1. Q1Integrate the following function: $\sin 2x$Free
  2. Q2Integrate the following function: $\cos 3x$Free
  3. Q3Find an anti derivative (or integral) of the function $e^{2x}$ by the method of inspectionFree
  4. Q4Integrate the following function: $(ax + b)^2$Preview
  5. Q5Find an anti derivative (or integral) of the function $\sin 2x - 4 e^{3x}$ by the method of inspection.Preview
  6. Q6Integrate the following function: $\int (4 e^{3x} + 1) dx$Preview
  7. Q7Integrate the following function: $\int x^2 \left(1 - \frac{1}{x^2}\right) dx$Preview
  8. Q8Integrate the following function: $\int (ax^2 + bx + c) dx$Preview
  9. Q9Integrate the following function: $\int (2x^2 + e^x) dx$Preview
  10. Q10Integrate the following function: $\int \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^2 dx$Preview
  11. Q11Integrate the following function: $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$Preview
  12. Q12Integrate the following function: $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$Preview
  13. Q13Integrate the following function: $\int \frac{x^3 - x^2 + x - 1}{x - 1} dx$Preview
  14. Q14Integrate the following function: $\int (1 - x) \sqrt{x} dx$Preview
  15. Q15Integrate the following function: $\int \sqrt{x}(3x^2 + 2x + 3) dx$Preview
  16. Q16Integrate the following function: $\int (2x - 3\cos x + e^x) dx$Preview
  17. Q17Integrate the following function: $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$Preview
  18. Q18Integrate the following function: $\int \sec x (\sec x + \tan x) dx$Preview
  19. Q19Integrate the following function: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x} dx$Preview
  20. Q20Find the integral $\int \frac{2-3\sin x}{\cos^2 x}\,dx$Preview
  21. Q21Integrate the following function: The anti derivative of $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)$ equals (A) $\frac{1}{3}x^{\frac{1}{3}}…Preview
  22. Q22If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$. Then $f(x)$ is (A) $x^4 + \frac{1}{x^3} + \frac{129}{8}$ (B) $x^3 + \fra…Preview
7.3

Methods of Integration

The method of inspection works for simple functions but quickly becomes impractical. To handle a wider variety of integrands, we use systematic techniques that transform unfamiliar integrals into stan…

7.3.1

Integration by Substitution

41 Q

Integration by substitution reverses the chain rule. It transforms a complicated integral into a simpler one by changing the variable of integration. For , set where is differentiable; then:

+Worked Examplesi2 questions
  1. Example 5Integrate the following functions w.r.t. $x$: (i) $\sin mx$ (ii) $2x \sin(x^2 + 1)$ (iii) $\dfrac{\tan^4 \sqrt{x}\,\sec^2 \sqrt{x}}{\sqrt{x}…Free
  2. Example 6Find the following integrals: (i) $\int \sin^3 x \cos^2 x\, dx$ (ii) $\int \dfrac{\sin x}{\sin(x + a)}\, dx$ (iii) $\int \dfrac{1}{1 + \tan…Preview
+Exercise 7.2i39 questions
  1. Q1Integrate the following function: $\frac{2x}{1+x^2}$Free
  2. Q2Integrate the following function: $\frac{(\log x)^2}{x}$Free
  3. Q3Integrate the following function: $\frac{1}{x+x \log x}$Free
  4. Q4Integrate the following function: $\sin x \sin (\cos x)$Preview
  5. Q5Integrate the following function: $\sin (ax+b) \cos (ax+b)$Preview
  6. Q6Integrate the function $\sqrt{ax+b}$Preview
  7. Q7Integrate the following function: $x \sqrt{x+2}$Preview
  8. Q8Integrate the following function: $x \sqrt{1+2x^2}$Preview
  9. Q9Integrate the following function: $(4x+2) \sqrt{x^2+x+1}$Preview
  10. Q10Integrate the function $\frac{1}{x-\sqrt{x}}$Preview
  11. Q11Integrate the following function: $\frac{x}{\sqrt{x+4}}$, $x>0$Preview
  12. Q12Integrate the following function: $(x^3-1)^{1/3} x^5$Preview
  13. Q13Integrate the following function: $\frac{x^2}{(2+3x^3)^3}$Preview
  14. Q14Integrate the following function: $\frac{1}{x (\log x)^m}$, $x>0, m \neq 1$Preview
  15. Q15Integrate the following function: $\frac{x}{9-4x^2}$Preview
  16. Q16Integrate the following function: $e^{2x+3}$Preview
  17. Q17Integrate the following function: $\frac{x}{e^{x^2}}$Preview
  18. Q18Integrate the following function: $\frac{e^{\tan^{-1} x}}{1+x^2}$Preview
  19. Q19Integrate the following function: $\frac{e^{2x}-1}{e^{2x}+1}$Preview
  20. Q20Integrate the following function: $\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$Preview
  21. Q21Integrate the following function: $ \tan^2 (2x - 3) $Preview
  22. Q22Integrate the following function: $ \sec^2 (7 - 4x) $Preview
  23. Q23Integrate the following function: $ \frac{\sin^{-1}x}{\sqrt{1 - x^2}} $Preview
  24. Q24Integrate the following function: $ \frac{2\cos x - 3\sin x}{6\cos x + 4\sin x} $Preview
  25. Q25Integrate the following function: $ \frac{1}{\cos^2 x (1 - \tan x)^2} $Preview
  26. Q26Integrate the following function: $ \frac{\cos \sqrt{x}}{\sqrt{x}} $Preview
  27. Q27Integrate the following function: $ \sqrt{\sin 2x} \cos 2x $Preview
  28. Q28Integrate the following function: $ \frac{\cos x}{\sqrt{1 + \sin x}} $Preview
  29. Q29Integrate the following function: $ \cot x \log \sin x $Preview
  30. Q30Integrate the following function: $ \frac{\sin x}{1 + \cos x} $Preview
  31. Q31Integrate the following function: $ \frac{\sin x}{(1 + \cos x)^2} $Preview
  32. Q32Integrate the following function: $ \frac{1}{1 + \cot x} $Preview
  33. Q33Integrate the following function: $ \frac{1}{1 - \tan x} $Preview
  34. Q34Integrate the following function: $ \frac{\sqrt{\tan x}}{\sin x \cos x} $Preview
  35. Q35Integrate the following function: $ \frac{(1 + \log x)^2}{x} $Preview
  36. Q36Integrate the following function: $ \frac{(x + 1)(x + \log x)^2}{x} $Preview
  37. Q37Integrate the function $\frac{x^3\sin(\tan^{-1}x^4)}{1+x^8}$Preview
  38. Q38Integrate the following function: $ \int \frac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x} dx $ equals (A) $ 10^x - x^{10} + C $ (B) $ 10^x + x^{…Preview
  39. Q39$ \int \frac{dx}{\sin^2 x \cos^2 x} $ equals (A) $ \tan x + \cot x + C $ (B) $ \tan x - \cot x + C $ (C) $ \tan x \cot x + C $ (D) $ \tan x…Preview
7.3.2

Integration Using Trigonometric Identities

25 Q

When the integrand is a power of a trigonometric function, or a product of sines and cosines, the standard integration formulas usually cannot be applied directly.

+Worked Examplesi1 question
  1. Example 7Find (i) $\int \cos^2 x\, dx$ (ii) $\int \sin 2x \cos 3x\, dx$ (iii) $\int \sin^3 x\, dx$Preview
+Exercise 7.3i24 questions
  1. Q1Integrate the following function: $\sin^2 (2x + 5)$Free
  2. Q2Integrate the following function: $\sin 3x \cos 4x$Free
  3. Q3Integrate the following function: $\cos 2x \cos 4x \cos 6x$Free
  4. Q4Integrate the following function: $\sin^3 (2x + 1)$Preview
  5. Q5Integrate the following function: $\sin^3 x \cos^3 x$Preview
  6. Q6Integrate the following function: $\sin x \sin 2x \sin 3x$Preview
  7. Q7Integrate the following function: $\sin 4x \sin 8x$Preview
  8. Q8Integrate the following function: $\frac{1 - \cos x}{1 + \cos x}$Preview
  9. Q9Integrate the following function: $\frac{\cos x}{1 + \cos x}$Preview
  10. Q10Integrate the following function: $\sin^4 x$Preview
  11. Q11Integrate the following function: $\cos^4 2x$Preview
  12. Q12Integrate the following function: $\frac{\sin^2 x}{1 + \cos x}$Preview
  13. Q13Integrate the following function: $\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}$Preview
  14. Q14Integrate the following function: $\frac{\cos x - \sin x}{1 + \sin 2x}$Preview
  15. Q15Integrate the following function: $\tan^3 2x \sec 2x$Preview
  16. Q16Integrate the following function: $\tan^4 x$Preview
  17. Q17Integrate the following function: $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$Preview
  18. Q18Integrate the following function: $\frac{\cos 2x + 2\sin^2 x}{\cos^2 x}$Preview
  19. Q19Integrate the following function: $\frac{1}{\sin x \cos^3 x}$Preview
  20. Q20Integrate the following function: $\frac{\cos 2x}{(\cos x + \sin x)^2}$Preview
  21. Q21Integrate the following function: $\sin^{-1} (\cos x)$Preview
  22. Q22Find the integral of the function $\frac{1}{\cos(x-a)\cos(x-b)}$Preview
  23. Q23$\int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx$ is equal to (A) $\tan x + \cot x + C$ (B) $\tan x + \operatorname{cosec} x + C$ (C)…Preview
  24. Q24Integrate the following function: $\int \frac{e^x (1 + x)}{\cos^2 (e^x x)} dx$ equals (A) $-\cot (e^x x) + C$ (B) $\tan (xe^x) + C$ (C) $\ta…Preview
7.4

Integrals of Some Particular Functions

28 Q

This section develops six fundamental integration formulae that serve as building blocks for a wider class of integrals.

+Worked Examplesi3 questions
  1. Example 8Find the following integrals: (i) $\int \dfrac{dx}{x^2 - 16}$ (ii) $\int \dfrac{dx}{\sqrt{2x - x^2}}$Free
  2. Example 9Find the following integrals: (i) $\int \dfrac{dx}{x^2 - 6x + 13}$ (ii) $\int \dfrac{dx}{3x^2 + 13x - 10}$ (iii) $\int \dfrac{dx}{\sqrt{5x^2…Preview
  3. Example 10Find the following integrals: (i) $\int \dfrac{x + 2}{2x^2 + 6x + 5}\, dx$ (ii) $\int \dfrac{x + 3}{\sqrt{5 - 4x - x^2}}\, dx$Preview
+Exercise 7.4i25 questions
  1. Q1Integrate the function $\frac{3x^2}{x^6+1}$Free
  2. Q2Integrate the following function: $\frac{1}{\sqrt{1+4x^2}}$Free
  3. Q3Integrate the following function: $\frac{1}{\sqrt{(2-x)^2+1}}$Free
  4. Q4Integrate the following function: $\frac{1}{\sqrt{9-25x^2}}$Preview
  5. Q5Integrate the following function: $\frac{3x}{1+2x^4}$Preview
  6. Q6Integrate the function $\frac{x^2}{1-x^6}$Preview
  7. Q7Integrate the following function: $\frac{x-1}{\sqrt{x^2-1}}$Preview
  8. Q8Integrate the following function: $\frac{x^2}{\sqrt{x^6+a^6}}$Preview
  9. Q9Integrate the following function: $\frac{\sec^2 x}{\sqrt{\tan^2 x+4}}$Preview
  10. Q10Integrate the following function: $\frac{1}{\sqrt{x^2 + 2x + 2}}$Preview
  11. Q11Integrate the following function: $\frac{1}{9x^2 + 6x + 5}$Preview
  12. Q12Integrate the following function: $\frac{1}{\sqrt{7 - 6x - x^2}}$Preview
  13. Q13Integrate the following function: $\frac{1}{\sqrt{(x-1)(x-2)}}$Preview
  14. Q14Integrate the following function: $\frac{1}{\sqrt{8 + 3x - x^2}}$Preview
  15. Q15Integrate the following function: $\frac{1}{\sqrt{(x-a)(x-b)}}$Preview
  16. Q16Integrate the following function: $\frac{4x+1}{\sqrt{2x^2 + x - 3}}$Preview
  17. Q17Integrate the following function: $\frac{x+2}{\sqrt{x^2 - 1}}$Preview
  18. Q18Integrate the following function: $\frac{5x-2}{1+2x+3x^2}$Preview
  19. Q19Integrate the following function: $\frac{6x+7}{\sqrt{(x-5)(x-4)}}$Preview
  20. Q20Integrate the following function: $\frac{x+2}{\sqrt{4x - x^2}}$Preview
  21. Q21Integrate the following function: $\frac{x+2}{\sqrt{x^2 + 2x + 3}}$Preview
  22. Q22Integrate the following function: $\frac{x+3}{x^2 - 2x - 5}$Preview
  23. Q23Integrate the function $\frac{5x+3}{\sqrt{x^2+4x+10}}$Preview
  24. Q24Integrate the following function: $\int \frac{dx}{x^2 + 2x + 2}$ equals (A) $x \tan^{-1} (x + 1) + C$ (B) $\tan^{-1} (x + 1) + C$ (C) $(x +…Preview
  25. Q25Integrate the following function: $\int \frac{dx}{\sqrt{9x - 4x^2}}$ equals (A) $\frac{1}{9} \sin^{-1} \left(\frac{9x - 8}{8}\right) + C$ (B…Preview
7.5

Integration by Partial Fractions

29 Q

A rational function is the ratio of two polynomials:

+Worked Examplesi6 questions
  1. Example 11Find $\int \dfrac{dx}{(x+1)(x+2)}$Free
  2. Example 12Find $\int \dfrac{x^2 + 1}{x^2 - 5x + 6}\, dx$Free
  3. Example 13Find $\int \dfrac{3x - 2}{(x+1)^2 (x+3)}\, dx$Preview
  4. Example 14Find $\int \dfrac{x^2}{(x^2+1)(x^2+4)}\, dx$Preview
  5. Example 15Find $\int \dfrac{(3\sin\phi - 2)\cos\phi}{5 - \cos^2\phi - 4\sin\phi}\, d\phi$Preview
  6. Example 16Find $\int \dfrac{x^2 + x + 1}{(x+2)(x^2+1)}\, dx$Preview
+Exercise 7.5i23 questions
  1. Q1Integrate the following function: $\frac{x}{(x + 1)(x + 2)}$Free
  2. Q2Integrate the following function: $\frac{1}{x^2 - 9}$Free
  3. Q3Integrate the following function: $\frac{3x - 1}{(x - 1)(x - 2)(x - 3)}$Free
  4. Q4Integrate the following function: $\frac{x}{(x - 1)(x - 2)(x - 3)}$Preview
  5. Q5Integrate the following function: $\frac{2x}{x^2 + 3x + 2}$Preview
  6. Q6Integrate the following function: $\frac{1 - x^2}{x(1 - 2x)}$Preview
  7. Q7Integrate the following function: $\frac{x}{(x^2 + 1)(x - 1)}$Preview
  8. Q8Integrate the following function: $\frac{x}{(x - 1)^2 (x + 2)}$Preview
  9. Q9Integrate the following function: $\frac{3x + 5}{x^3 - x^2 - x + 1}$Preview
  10. Q10Integrate the following function: $\frac{2x - 3}{(x^2 - 1)(2x + 3)}$Preview
  11. Q11Integrate the following function: $\frac{5x}{(x + 1)(x^2 - 4)}$Preview
  12. Q12Integrate the following function: $\frac{x^3 + x + 1}{x^2 - 1}$Preview
  13. Q13Integrate the following function: $\frac{2}{(1 - x)(1 + x^2)}$Preview
  14. Q14Integrate the following function: $\frac{3x - 1}{(x + 2)^2}$Preview
  15. Q15Integrate the following function: $\frac{1}{x^4 - 1}$Preview
  16. Q16Integrate the following function: $\frac{1}{x(x^n + 1)}$ [Hint: multiply numerator and denominator by $x^{n-1}$ and put $x^n = t$]Preview
  17. Q17Integrate the following function: $\frac{\cos x}{(1 - \sin x)(2 - \sin x)}$ [Hint : Put $\sin x = t$]Preview
  18. Q18Integrate the following function: $\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$Preview
  19. Q19Integrate the following function: $\frac{2x}{(x^2+1)(x^2+3)}$Preview
  20. Q20Integrate the following function: $\frac{1}{x(x^4-1)}$Preview
  21. Q21Integrate the rational function $\frac{1}{e^x-1}$ [Hint: Put $e^x=t$]Preview
  22. Q22Integrate the following function: $\int \frac{x \ dx}{(x-1)(x-2)}$ equals (A) $\log \left| \frac{(x-1)^2}{x-2} \right| + \text{C}$ (B) $\log…Preview
  23. Q23Integrate the following function: $\int \frac{dx}{x(x^2+1)}$ equals (A) $\log |x| - \frac{1}{2} \log (x^2+1) + \text{C}$ (B) $\log |x| + \fr…Preview
7.6

Integration by Parts

Integration by parts integrates products of functions. It is derived directly from the product rule of differentiation and transforms a difficult integral into a simpler one.

+Worked Examplesi5 questions
  1. Example 17Find $\int x \cos x\, dx$Free
  2. Example 18Find $\int \log x\, dx$Free
  3. Example 19Find $\int x\, e^x\, dx$Preview
  4. Example 20Find $\int \dfrac{x \sin^{-1} x}{\sqrt{1 - x^2}}\, dx$Preview
  5. Example 21Find $\int e^x \sin x\, dx$Preview
7.6.1

Integral of the Type

24 Q

This section develops a shortcut for integrating expressions where multiplies the sum of a function and its derivative.

+Worked Examplesi1 question
  1. Example 22Find (i) $\int e^x\left(\tan^{-1} x + \dfrac{1}{1 + x^2}\right) dx$ (ii) $\int \dfrac{(x^2 + 1)\,e^x}{(x+1)^2}\, dx$Preview
+Exercise 7.6i23 questions
  1. Q1Integrate the following function: $x \sin x$Free
  2. Q2Integrate the following function: $x \sin 3x$Free
  3. Q3Integrate the function $x^2 e^x$Free
  4. Q4Integrate the following function: $x \log x$Preview
  5. Q5Integrate the following function: $x \log 2x$Preview
  6. Q6Integrate the following function: $x^2 \log x$Preview
  7. Q7Integrate the following function: $x \sin^{-1}x$Preview
  8. Q8Integrate the following function: $\tan^{-1}x$Preview
  9. Q9Integrate the following function: $x \cos^{-1}x$Preview
  10. Q10Integrate the following function: $(\sin^{-1}x)^2$Preview
  11. Q11Integrate the following function: $\frac{x \cos^{-1}x}{\sqrt{1-x^2}}$Preview
  12. Q12Integrate the following function: $x \sec^2 x$Preview
  13. Q14Integrate the following function: $x (\log x)^2$Preview
  14. Q15Integrate the following function: $(x^2+1) \log x$Preview
  15. Q16Integrate the following function: $e^x (\sin x + \cos x)$Preview
  16. Q17Integrate the following function: $\frac{x e^x}{(1+x)^2}$Preview
  17. Q18Integrate the following function: $e^x \left(\frac{1+\sin x}{1+\cos x}\right)$Preview
  18. Q19Integrate the following function: $e^x \left(\frac{1}{x} - \frac{1}{x^2}\right)$Preview
  19. Q20Integrate the following function: $\frac{(x-3)e^x}{(x-1)^3}$Preview
  20. Q21Integrate the following function: $e^{2x} \sin x$Preview
  21. Q22Integrate the function $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$Preview
  22. Q23Integrate the following function: $\int x^2 e^{x^3} dx$ equals (A) $\frac{1}{3} e^{x^3} + C$ (B) $\frac{1}{3} e^{x^2} + C$ (C) $\frac{1}{2}…Preview
  23. Q24$\int e^x \sec x\,(1+\tan x)\, dx$ equals (A) $e^x \cos x + C$ (B) $e^x \sec x + C$ (C) $e^x \sin x + C$ (D) $e^x \tan x + C$Preview
7.6.2

Integrals of Some More Types

13 Q

Integration by parts, with the constant function taken as the second function, yields three important standard integrals.

+Worked Examplesi2 questions
  1. Example 23Find $\int \sqrt{x^2 + 2x + 5}\, dx$Free
  2. Example 24Find $\int \sqrt{3 - 2x - x^2}\, dx$Preview
+Exercise 7.7i11 questions
  1. Q1Integrate the function $\sqrt{4-x^2}$Free
  2. Q2Integrate the following function: $\sqrt{1-4x^2}$Free
  3. Q3Integrate the following function: $\sqrt{x^2+4x+6}$Free
  4. Q4Integrate the following function: $\sqrt{x^2+4x+1}$Preview
  5. Q5Integrate the following function: $\sqrt{1-4x-x^2}$Preview
  6. Q6Integrate the following function: $\sqrt{x^2+4x-5}$Preview
  7. Q7Integrate the following function: $\sqrt{1+3x-x^2}$Preview
  8. Q8Integrate the following function: $\sqrt{x^2+3x}$Preview
  9. Q9Integrate the function $\sqrt{1+\frac{x^2}{9}}$Preview
  10. Q10$\int \sqrt{1+x^2} dx$ is equal to (A) $\frac{x}{2}\sqrt{1+x^2} + \frac{1}{2}\log |x+\sqrt{1+x^2}| + C$ (B) $\frac{2}{3}(1+x^2)^{\frac{3}{2}…Preview
  11. Q11$\int \sqrt{x^2-8x+7} dx$ is equal to (A) $\frac{1}{2}(x-4)\sqrt{x^2-8x+7} + 9\log |x-4+\sqrt{x^2-8x+7}| + C$ (B) $\frac{1}{2}(x-4)\sqrt{x^2…Preview
7.7

Definite Integral

Earlier you met indefinite integrals — families of functions that differ by a constant. The definite integral instead has a single, unique numerical value; it is a number, not a family of functions.

7.8

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) reveals the deep connection between differentiation (finding slopes/rates of change) and integration (finding areas/accumulated change): the two operations ar…

7.8.1

Area Function

The definite integral is the area of the region bounded by the curve , the -axis, and the vertical lines and — a fixed number for given and . But what happens if we let the upper limit vary?

7.8.2

First Fundamental Theorem of Integral Calculus

This theorem establishes the crucial link between the two branches of calculus: differentiation and integration.

7.8.3

Second Fundamental Theorem of Integral Calculus

23 Q

The Second Fundamental Theorem of Integral Calculus makes evaluating definite integrals practical. Instead of calculating limits of sums, it lets you use an antiderivative (indefinite integral) to fin…

+Worked Examplesi1 question
  1. Example 25Evaluate the following integrals: (i) $\int_2^3 x^2\, dx$ (ii) $\int_4^9 \dfrac{\sqrt{x}}{(30 - x^{3/2})^2}\, dx$ (iii) $\int_1^2 \dfrac{x\,…Preview
+Exercise 7.8i22 questions
  1. Q1Evaluate the definite integral: $\int_{-1}^1 (x+1) \ dx$Free
  2. Q2Evaluate the definite integral: $\int_2^3 \frac{1}{x} \ dx$Free
  3. Q3Evaluate the definite integral: $\int_1^2 (4x^3 - 5x^2 + 6x + 9) \ dx$Free
  4. Q4Evaluate the definite integral: $\int_0^{\pi/4} \sin 2x \ dx$Preview
  5. Q5Evaluate the definite integral: $\int_0^{\pi/2} \cos 2x \ dx$Preview
  6. Q6Evaluate the definite integral: $\int_4^5 e^x \ dx$Preview
  7. Q7Evaluate the definite integral: $\int_0^{\pi/4} \tan x \ dx$Preview
  8. Q8Evaluate the definite integral: $\int_{\pi/6}^{\pi/4} \operatorname{cosec} x \ dx$Preview
  9. Q9Evaluate the definite integral: $\int_0^1 \frac{dx}{\sqrt{1-x^2}}$Preview
  10. Q10Evaluate the definite integral: $\int_0^1 \frac{dx}{1+x^2}$Preview
  11. Q11Evaluate the definite integral: $\int_2^3 \frac{dx}{x^2-1}$Preview
  12. Q12Evaluate the definite integral: $\int_{0}^{\pi} \cos^2 x \, dx$Preview
  13. Q13Evaluate the definite integral: $\int_{2}^{3} \frac{x \, dx}{x^2+1}$Preview
  14. Q14Evaluate the definite integral: $\int_{0}^{1} \frac{2x+3}{5x^2+1} \, dx$Preview
  15. Q15Evaluate the definite integral: $\int_{0}^{1} x e^{x^2} \, dx$Preview
  16. Q16Evaluate the definite integral: $\int_{1}^{2} \frac{5x^2}{x^2+4x+3} \, dx$Preview
  17. Q17Evaluate the definite integral: $\int_{0}^{\pi/4} (2 \sec^2 x + x^3 + 2) \, dx$Preview
  18. Q18Evaluate the definite integral: $\int_{0}^{\pi} \left(\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}\right) \, dx$Preview
  19. Q19Evaluate the definite integral: $\int_{0}^{2} \frac{6x+3}{x^2+4} \, dx$Preview
  20. Q20Evaluate the definite integral $\int_{0}^{1}\left(x\,e^x+\sin\frac{\pi x}{4}\right)dx$Preview
  21. Q21Evaluate the definite integral: $\int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}$ equals (A) $\frac{\pi}{3}$ (B) $\frac{2\pi}{3}$ (C) $\frac{\pi}{6}$ (…Preview
  22. Q22Evaluate the definite integral: $\int_{0}^{2/3} \frac{dx}{4+9x^2}$ equals (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{12}$ (C) $\frac{\pi}{24}$ (D)…Preview
7.9

Evaluation of Definite Integrals by Substitution

12 Q

The method of substitution, used extensively for indefinite integrals, adapts to evaluate definite integrals directly.

+Worked Examplesi2 questions
  1. Example 26Evaluate $\int_{-1}^1 5x^4 \sqrt{x^5 + 1}\, dx$Free
  2. Example 27Evaluate $\int_0^1 \dfrac{\tan^{-1} x}{1 + x^2}\, dx$Preview
+Exercise 7.9i10 questions
  1. Q1Evaluate the integral using substitution $\int_{0}^{1}\frac{x}{x^2+1}\,dx$Free
  2. Q2Evaluate the integral using substitution $\int_{0}^{\pi/2}\sqrt{\sin\phi}\,\cos^5\phi\,d\phi$Free
  3. Q3Evaluate the integral using substitution $\int_{0}^{1}\sin^{-1}\left(\frac{2x}{1+x^2}\right)dx$Free
  4. Q4Evaluate the integral using substitution $\int_{0}^{2}x\sqrt{x+2}\,dx$ (Put $x+2=t^2$)Preview
  5. Q5Evaluate the integral using substitution $\int_{0}^{\pi/2}\frac{\sin x}{1+\cos^2 x}\,dx$Preview
  6. Q6Evaluate the integral using substitution $\int_{0}^{2}\frac{dx}{x+4-x^2}$Preview
  7. Q7Evaluate the integral using substitution $\int_{-1}^{1}\frac{dx}{x^2+2x+5}$Preview
  8. Q8Evaluate the integral using substitution $\int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2x^2}\right)e^{2x}\,dx$Preview
  9. Q9Choose the correct answer: The value of the integral $\int_{1/3}^{1}\frac{(x-x^3)^{1/3}}{x^4}\,dx$ is (A) 6 (B) 0 (C) 3 (D) 4Preview
  10. Q10Choose the correct answer: If $f(x)=\int_{0}^{x}t\sin t\,dt$, then $f'(x)$ is (A) $\cos x+x\sin x$ (B) $x\sin x$ (C) $x\cos x$ (D) $\sin x+x…Preview
7.10

Some Properties of Definite Integrals

28 Q

Definite integrals can often be evaluated more easily using properties that relate integrals over different intervals or with transformed integrands.

+Worked Examplesi7 questions
  1. Example 28Evaluate $\int_{-1}^2 \left| x^3 - x \right|\, dx$Free
  2. Example 29Evaluate $\int_{-\pi/4}^{\pi/4} \sin^2 x\, dx$Free
  3. Example 30Evaluate $\int_0^{\pi} \dfrac{x \sin x}{1 + \cos^2 x}\, dx$Free
  4. Example 31Evaluate $\int_{-1}^1 \sin^5 x \cos^4 x\, dx$Preview
  5. Example 32Evaluate $\int_0^{\pi/2} \dfrac{\sin^4 x}{\sin^4 x + \cos^4 x}\, dx$Preview
  6. Example 33Evaluate $\int_{\pi/6}^{\pi/3} \dfrac{dx}{1 + \sqrt{\tan x}}$Preview
  7. Example 34Evaluate $\int_0^{\pi/2} \log \sin x\, dx$Preview
+Exercise 7.10i21 questions
  1. Q1By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}\cos^2 x\,dx$Free
  2. Q2By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$Free
  3. Q3By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$Free
  4. Q4By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}\frac{\cos^5 x}{\sin^5 x+\cos^5 x}\,dx$Preview
  5. Q5By using the properties of definite integrals, evaluate the integral $\int_{-5}^{5}|x+2|\,dx$Preview
  6. Q6By using the properties of definite integrals, evaluate the integral $\int_{2}^{8}|x-5|\,dx$Preview
  7. Q7By using the properties of definite integrals, evaluate the integral $\int_{0}^{1}x(1-x)^n\,dx$Preview
  8. Q8By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/4}\log(1+\tan x)\,dx$Preview
  9. Q9By using the properties of definite integrals, evaluate the integral $\int_{0}^{2}x\sqrt{2-x}\,dx$Preview
  10. Q10By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}(2\log\sin x-\log\sin 2x)\,dx$Preview
  11. Q11By using the properties of definite integrals, evaluate the integral $\int_{-\pi/2}^{\pi/2}\sin^2 x\,dx$Preview
  12. Q12By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi}\frac{x\,dx}{1+\sin x}$Preview
  13. Q13By using the properties of definite integrals, evaluate the integral $\int_{-\pi/2}^{\pi/2}\sin^7 x\,dx$Preview
  14. Q14By using the properties of definite integrals, evaluate the integral $\int_{0}^{2\pi}\cos^5 x\,dx$Preview
  15. Q15By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi/2}\frac{\sin x-\cos x}{1+\sin x\cos x}\,dx$Preview
  16. Q16By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi}\log(1+\cos x)\,dx$Preview
  17. Q17By using the properties of definite integrals, evaluate the integral $\int_{0}^{a}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}\,dx$Preview
  18. Q18By using the properties of definite integrals, evaluate the integral $\int_{0}^{4}|x-1|\,dx$Preview
  19. Q19Show that $\int_{0}^{a}f(x)g(x)\,dx=2\int_{0}^{a}f(x)\,dx$, if $f$ and $g$ are defined as $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$Preview
  20. Q20Choose the correct answer: The value of $\int_{-\pi/2}^{\pi/2}(x^3+x\cos x+\tan^5 x+1)\,dx$ is (A) 0 (B) 2 (C) $\pi$ (D) 1Preview
  21. Q21Choose the correct answer: The value of $\int_{0}^{\pi/2}\log\left(\frac{4+3\sin x}{4+3\cos x}\right)dx$ is (A) 2 (B) $\frac{3}{4}$ (C) 0 (D…Preview

Miscellaneous Examples

+Miscellaneous Examplesi8 questions
  1. Example 35Find $\int \cos 6x \sqrt{1 + \sin 6x}\, dx$Free
  2. Example 36Find $\int \dfrac{(x^4 - x)^{1/4}}{x^5}\, dx$Free
  3. Example 37Find $\int \dfrac{x^4\, dx}{(x-1)(x^2+1)}$Free
  4. Example 38Find $\int \left[\log(\log x) + \dfrac{1}{(\log x)^2}\right] dx$Preview
  5. Example 39Find $\int \left[\sqrt{\cot x} + \sqrt{\tan x}\right] dx$Preview
  6. Example 40Find $\int \dfrac{\sin 2x \cos 2x\, dx}{\sqrt{9 - \cos^4(2x)}}$Preview
  7. Example 41Evaluate $\int_{-1}^{3/2} \left| x \sin(\pi x) \right|\, dx$Preview
  8. Example 42Evaluate $\int_0^{\pi} \dfrac{x\, dx}{a^2 \cos^2 x + b^2 \sin^2 x}$Preview

Miscellaneous Exercise on Chapter 7

+Miscellaneous Exercisei40 questions
  1. Q1Integrate the function $\frac{1}{x-x^3}$Free
  2. Q2Integrate the function $\frac{1}{\sqrt{x+a}+\sqrt{x+b}}$Free
  3. Q3Integrate the function $\frac{1}{x\sqrt{ax-x^2}}\ \text{[Hint: Put } x=\frac{a}{t}\text{]}$Free
  4. Q4Integrate the function $\frac{1}{x^2(x^4+1)^{3/4}}$Preview
  5. Q5Integrate the function: $\displaystyle \int \frac{1}{x^{1/2}+x^{1/3}}\,dx$ [Hint: put $x=t^{6}$]Preview
  6. Q6Integrate the function $\frac{5x}{(x+1)(x^2+9)}$Preview
  7. Q7Integrate the function $\frac{\sin x}{\sin(x-a)}$Preview
  8. Q8Integrate the function $\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}$Preview
  9. Q9Integrate the function $\frac{\cos x}{\sqrt{4-\sin^2 x}}$Preview
  10. Q10Integrate the function $\frac{\sin^8 x-\cos^8 x}{1-2\sin^2 x\cos^2 x}$Preview
  11. Q11Integrate the function $\frac{1}{\cos(x+a)\cos(x+b)}$Preview
  12. Q12Integrate the function $\frac{x^3}{\sqrt{1-x^8}}$Preview
  13. Q13Integrate the function $\frac{e^x}{(1+e^x)(2+e^x)}$Preview
  14. Q14Integrate the function $\frac{1}{(x^2+1)(x^2+4)}$Preview
  15. Q15Integrate the function $\cos^3 x\,e^{\log\sin x}$Preview
  16. Q16Integrate the function $e^{3\log x}(x^4+1)^{-1}$Preview
  17. Q17Integrate the function $f'(ax+b)\,[f(ax+b)]^n$Preview
  18. Q18Integrate the function $\frac{1}{\sqrt{\sin^3 x\,\sin(x+\alpha)}}$Preview
  19. Q19Integrate the function $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$Preview
  20. Q20Integrate the function $\frac{2+\sin 2x}{1+\cos 2x}\,e^x$Preview
  21. Q21Integrate the function $\frac{x^2+x+1}{(x+1)^2(x+2)}$Preview
  22. Q22Integrate the function $\tan^{-1}\sqrt{\frac{1-x}{1+x}}$Preview
  23. Q23Integrate the function $\frac{\sqrt{x^2+1}\,[\log(x^2+1)-2\log x]}{x^4}$Preview
  24. Q24Evaluate the definite integral $\int_{\pi/2}^{\pi}e^x\left(\frac{1-\sin x}{1-\cos x}\right)dx$Preview
  25. Q25Evaluate the definite integral $\int_{0}^{\pi/4}\frac{\sin x\cos x}{\cos^4 x+\sin^4 x}\,dx$Preview
  26. Q26Evaluate the definite integral $\int_{0}^{\pi/2}\frac{\cos^2 x\,dx}{\cos^2 x+4\sin^2 x}$Preview
  27. Q27Evaluate the definite integral $\int_{\pi/6}^{\pi/3}\frac{\sin x+\cos x}{\sqrt{\sin 2x}}\,dx$Preview
  28. Q28Evaluate the definite integral $\int_{0}^{1}\frac{dx}{\sqrt{1+x}-\sqrt{x}}$Preview
  29. Q29Evaluate the definite integral $\int_{0}^{\pi/4}\frac{\sin x+\cos x}{9+16\sin 2x}\,dx$Preview
  30. Q30Evaluate the definite integral $\int_{0}^{\pi/2}\sin 2x\,\tan^{-1}(\sin x)\,dx$Preview
  31. Q31Evaluate the definite integral $\int_{1}^{4}\left[|x-1|+|x-2|+|x-3|\right]dx$Preview
  32. Q32Prove that $\int_{1}^{3}\frac{dx}{x^2(x+1)}=\frac{2}{3}+\log\frac{2}{3}$Preview
  33. Q33Prove that $\int_{0}^{1}x\,e^x\,dx=1$Preview
  34. Q34Prove that $\int_{-1}^{1}x^{17}\cos^4 x\,dx=0$Preview
  35. Q35Prove that $\int_{0}^{\pi/2}\sin^3 x\,dx=\frac{2}{3}$Preview
  36. Q36Prove that $\int_{0}^{\pi/4}2\tan^3 x\,dx=1-\log 2$Preview
  37. Q37Prove that $\int_{0}^{1}\sin^{-1}x\,dx=\frac{\pi}{2}-1$Preview
  38. Q38Choose the correct answer: $\int \frac{dx}{e^x+e^{-x}}$ is equal to (A) $\tan^{-1}(e^x)+C$ (B) $\tan^{-1}(e^{-x})+C$ (C) $\log(e^x-e^{-x})+C…Preview
  39. Q39Choose the correct answer: $\int \frac{\cos 2x}{(\sin x+\cos x)^2}\,dx$ is equal to (A) $\frac{-1}{\sin x+\cos x}+C$ (B) $\log|\sin x+\cos x…Preview
  40. Q40Choose the correct answer: If $f(a+b-x)=f(x)$, then $\int_{a}^{b}x\,f(x)\,dx$ is equal to (A) $\frac{a+b}{2}\int_{a}^{b}f(b-x)\,dx$ (B) $\fr…Preview

Summary

- Integration as anti-derivative: If , then , where is the constant of integration. - Basic formulas: Memorise standard integrals like (), , , and trigonometric integrals (, etc.).

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 63 questions63 questions
  1. Q1Verify: $\int \dfrac{2x-1}{2x+3}\,dx = x - \log|(2x+3)^2| + C$Free
  2. Q2Verify: $\int \dfrac{2x+3}{x^2+3x}\,dx = \log|x^2+3x| + C$Free
  3. Q3Evaluate: $\int \dfrac{x^2+2}{x+1}\,dx$Free
  4. Q4Evaluate: $\int \dfrac{e^{6\log x}-e^{5\log x}}{e^{4\log x}-e^{3\log x}}\,dx$Preview
  5. Q5Evaluate: $\int \dfrac{1+\cos x}{x+\sin x}\,dx$Preview
  6. Q6Evaluate: $\int \dfrac{dx}{1+\cos x}$Preview
  7. Q7Evaluate: $\int \tan^2 x\,\sec^4 x\,dx$Preview
  8. Q8Evaluate: $\int \dfrac{\sin x+\cos x}{\sqrt{1+\sin 2x}}\,dx$Preview
  9. Q9Evaluate: $\int \sqrt{1+\sin x}\,dx$Preview
  10. Q10Evaluate: $\int \dfrac{x}{1+\sqrt{x}}\,dx$ (Hint: Put $\sqrt{x}=z$)Preview
  11. Q11Evaluate: $\int \sqrt{\dfrac{a+x}{a-x}}\,dx$Preview
  12. Q12Evaluate: $\int \dfrac{x^{1/2}}{1+x^{3/4}}\,dx$ (Hint: Put $x=z^4$)Preview
  13. Q13Evaluate: $\int \dfrac{\sqrt{1+x^2}}{x^4}\,dx$Preview
  14. Q14Evaluate: $\int \dfrac{dx}{\sqrt{16-9x^2}}$Preview
  15. Q15Evaluate: $\int \dfrac{dt}{\sqrt{3-2t-t^2}}$Preview
  16. Q16Evaluate: $\int \dfrac{3x-1}{\sqrt{x^2+9}}\,dx$Preview
  17. Q17Evaluate: $\int \sqrt{5-2x+x^2}\,dx$Preview
  18. Q18Evaluate: $\int \dfrac{x}{x^4-1}\,dx$Preview
  19. Q19Evaluate: $\int \dfrac{x^2}{1-x^4}\,dx$ (put $x^2=t$)Preview
  20. Q20Evaluate: $\int \sqrt{2ax-x^2}\,dx$Preview
  21. Q21Evaluate: $\int \dfrac{\sin^{-1}x}{(1-x^2)^{3/2}}\,dx$Preview
  22. Q22Evaluate: $\int \dfrac{\cos 5x+\cos 4x}{1-2\cos 3x}\,dx$Preview
  23. Q23Evaluate: $\int \dfrac{\sin^6 x+\cos^6 x}{\sin^2 x\,\cos^2 x}\,dx$Preview
  24. Q24Evaluate: $\int \dfrac{\sqrt{x}}{\sqrt{a^3-x^3}}\,dx$Preview
  25. Q25Evaluate: $\int \dfrac{\cos x-\cos 2x}{1-\cos x}\,dx$Preview
  26. Q26Evaluate: $\int \dfrac{dx}{x\sqrt{x^4-1}}$ (Hint: Put $x^2=\sec\theta$)Preview
  27. Q27Evaluate as a limit of sums: $\int_{0}^{2} (x^2+3)\,dx$Preview
  28. Q28Evaluate as a limit of sums: $\int_{0}^{2} e^{x}\,dx$Preview
  29. Q29Evaluate: $\int_{0}^{1} \dfrac{dx}{e^{x}+e^{-x}}$Preview
  30. Q30Evaluate: $\int_{0}^{\pi/2} \dfrac{\tan x}{1+m^2\tan^2 x}\,dx$Preview
  31. Q31Evaluate: $\int_{1}^{2} \dfrac{dx}{\sqrt{(x-1)(2-x)}}$Preview
  32. Q32Evaluate: $\int_{0}^{1} \dfrac{x\,dx}{\sqrt{1+x^2}}$Preview
  33. Q33Evaluate: $\int_{0}^{\pi} x\sin x\cos^2 x\,dx$Preview
  34. Q34Evaluate: $\int_{0}^{1} \dfrac{dx}{(1+x^2)\sqrt{1-x^2}}$ (Hint: let $x=\sin\theta$)Preview
  35. Q35$\int \dfrac{x^2\,dx}{x^4-x^2-12}$Preview
  36. Q36$\int \dfrac{x^2\,dx}{(x^2+a^2)(x^2+b^2)}$Preview
  37. Q37$\int_{0}^{\pi} \dfrac{x}{1+\sin x}\,dx$Preview
  38. Q38$\int \dfrac{2x-1}{(x-1)(x+2)(x-3)}\,dx$Preview
  39. Q39$\int e^{\tan^{-1}x}\left(\dfrac{1+x+x^2}{1+x^2}\right)dx$Preview
  40. Q40$\int \sin^{-1}\sqrt{\dfrac{x}{a+x}}\,dx$ (Hint: Put $x=a\tan^2\theta$)Preview
  41. Q41$\int_{\pi/3}^{\pi/2} \dfrac{\sqrt{1+\cos x}}{(1-\cos x)^{5/2}}\,dx$Preview
  42. Q42$\int e^{-3x}\cos^3 x\,dx$Preview
  43. Q43$\int \sqrt{\tan x}\,dx$ (Hint: Put $\tan x=t^2$)Preview
  44. Q44$\int_{0}^{\pi/2} \dfrac{dx}{(a^2\cos^2 x+b^2\sin^2 x)^2}$ (Hint: Divide numerator and denominator by $\cos^4 x$)Preview
  45. Q45$\int_{0}^{1} x\log(1+2x)\,dx$Preview
  46. Q46$\int_{0}^{\pi} x\log\sin x\,dx$Preview
  47. Q47$\int_{-\pi/4}^{\pi/4} \log(\sin x+\cos x)\,dx$Preview
  48. Q48$\int_{0}^{\pi/2} \cos x\,e^{\sin x}\,dx = $ _______.Preview
  49. Q49$\int \dfrac{x+3}{(x+4)^2}\,e^{x}\,dx = $ _______.Preview
  50. Q50If $\int_{0}^{a} \dfrac{dx}{1+4x^2} = \dfrac{\pi}{8}$, then $a = $ _______.Preview
  51. Q51$\int \dfrac{\sin x}{3+4\cos^2 x}\,dx = $ _______.Preview
  52. Q52The value of $\int_{-\pi}^{\pi} \sin 3x\,\cos 2x\,dx$ is _______.Preview
  53. Q53$\int \dfrac{\cos 2x-\cos 2\theta}{\cos x-\cos\theta}\,dx$ is equal to (A) $2(\sin x + x\cos\theta) + C$ (B) $2(\sin x - x\cos\theta) + C$ (…Preview
  54. Q54$\int \dfrac{dx}{\sin(x-a)\sin(x-b)}$ is equal to (A) $\sin(b-a)\,\log\left|\dfrac{\sin(x-b)}{\sin(x-a)}\right| + C$ (B) $\operatorname{cose…Preview
  55. Q55$\int \tan^{-1}\sqrt{x}\,dx$ is equal to (A) $(x+1)\tan^{-1}\sqrt{x} - \sqrt{x} + C$ (B) $x\tan^{-1}\sqrt{x} - \sqrt{x} + C$ (C) $\sqrt{x} -…Preview
  56. Q56$\int e^{x}\left(\dfrac{1-x}{1+x^2}\right)^2 dx$ is equal to (A) $\dfrac{e^{x}}{1+x^2} + C$ (B) $-\dfrac{e^{x}}{1+x^2} + C$ (C) $\dfrac{e^{x…Preview
  57. Q57$\int \dfrac{x^9}{(4x^2+1)^6}\,dx$ is equal to (A) $\dfrac{1}{5x}\left(\dfrac{1}{x^2}+4\right)^{-5} + C$ (B) $\dfrac{1}{5}\left(\dfrac{1}{x^…Preview
  58. Q58If $\int \dfrac{dx}{(x+2)(x^2+1)} = a\log|1+x^2| + b\tan^{-1}x + \dfrac{1}{5}\log|x+2| + C$, then (A) $a=-\dfrac{1}{10},\ b=-\dfrac{2}{5}$ (…Preview
  59. Q59$\int \dfrac{x^3}{x+1}\,dx$ is equal to (A) $x + \dfrac{x^2}{2} + \dfrac{x^3}{3} - \log|1-x| + C$ (B) $x + \dfrac{x^2}{2} - \dfrac{x^3}{3} -…Preview
  60. Q60$\int \dfrac{x+\sin x}{1+\cos x}\,dx$ is equal to (A) $\log|1+\cos x| + C$ (B) $\log|x+\sin x| + C$ (C) $x - \tan\dfrac{x}{2} + C$ (D) $x\ta…Preview
  61. Q61If $\int \dfrac{x^3}{\sqrt{1+x^2}}\,dx = a(1+x^2)^{3/2} + b\sqrt{1+x^2} + C$, then (A) $a=\dfrac{1}{3},\ b=1$ (B) $a=-\dfrac{1}{3},\ b=1$ (C…Preview
  62. Q62$\int_{-\pi/4}^{\pi/4} \dfrac{dx}{1+\cos 2x}$ is equal to (A) $1$ (B) $2$ (C) $3$ (D) $4$Preview
  63. Q63$\int_{0}^{\pi/2} \sqrt{1-\sin 2x}\,dx$ is equal to (A) $2\sqrt{2}$ (B) $2(\sqrt{2}+1)$ (C) $2$ (D) $2(\sqrt{2}-1)$Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 8 questions8 questions
  1. Q1If 𝑓(𝑎 + 𝑏 − 𝑥) = 𝑓(𝑥), then ∫ 𝑥 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 is equal to (A) 𝑎+𝑏 2 ∫ 𝑓(𝑏 − 𝑥)𝑑𝑥 𝑏 𝑎 (B) 𝑎+𝑏 2 ∫ 𝑓(𝑎 − 𝑥)𝑑𝑥 𝑏 𝑎 (C) 𝑏−𝑎 2 ∫ 𝑓(𝑥) 𝑑𝑥 𝑏 𝑎 (D) 𝑎+…Preview
  2. Q2If ∫ 𝑥3 𝑠𝑖𝑛4(𝑥4) cos(𝑥4) 𝑑𝑥 = 𝑎 𝑠𝑖𝑛5(𝑥4) + C, then 𝑎 is equal to (A) − 1 10 (B) 1 20 (C) 1 4 (D) 1 5Preview
  3. Q3∫ 𝒅𝒙 𝒙𝟑(𝟏+𝒙𝟒) 𝟏 𝟐 equals (A) − 1 2𝑥2 √1 + 𝑥4 + 𝑐 (B) 1 2𝑥 √1 + 𝑥4 + 𝑐 (C) − 1 4𝑥 √1 + 𝑥4 + 𝑐 (D) 1 4𝑥2 √1 + 𝑥4 + 𝑐Preview
  4. Q4$\int_{0}^{2\pi}\operatorname{cosec}^{7}x\,dx=$ (A) $0$ (B) $1$ (C) $4$ (D) $2\pi$Preview
  5. Q5A student observes an open-air Honeybee nest on the branch of a tree, whose plane figure is parabolic shape given by $x^{2}=4y$. Then the ar…Preview
  6. Q6Evaluate : $$\int \dfrac{\cos 2x + 2\sin^2 x}{\cos^2 x}\, dx$$Preview
  7. Q7Find : $$\int \dfrac{2\cos x}{(1 - \sin x)(1 + \sin^2 x)}\, dx$$Preview
  8. Q8Evaluate : $$\int_{0}^{\pi/4} \dfrac{\sin x + \cos x}{16 + 9\sin 2x}\, dx$$ **OR** Evaluate : $$\int_{1}^{3} (x^2 + 3x + e^x)\, dx$$ as the…Preview