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Mathematics · Class 12 Science

Ch 11Definite Integration — Class 12 Mathematics, concept-first.

A definite integral of a function on an interval is written , read as "the integral from to of with respect to ." Here are real numbers and is defined (and, for this introduction, continuous and non-negative) on .

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Definite Integral as a Limit of a Sum

Many limits of large sums are evaluated by recognising them as Riemann sums. If a sum can be written as , then because and as the partition is refined.

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4.1

Definite Integral as Limit of a Sum

A definite integral of a function on an interval is written , read as "the integral from to of with respect to ." Here are real numbers and is defined (and, for this introduction, continuous and non-n…

+Exercise 4.1i5 questions
  1. Q1Evaluate the following integral as limit of sum: $\int_1^3 (3x-4)\,dx$Free
  2. Q2Evaluate the following integral as limit of sum: $\int_0^4 x^2\,dx$Free
  3. Q3Evaluate the following integral as limit of sum: $\int_0^2 e^x\,dx$Preview
  4. Q4Evaluate the following integral as limit of sum: $\int_0^2 (3x^2-1)\,dx$Preview
  5. Q5Evaluate the following integral as limit of sum: $\int_1^3 x^3\,dx$Preview
4.2

Fundamental Theorem of Integral Calculus

Building a Riemann-type sum and taking a limit every time an integral needs to be evaluated (Section 4.1) is reliable but slow. The Fundamental Theorem of Integral Calculus removes that need entirely.

+Exercise 4.2i45 questions
  1. Q6Evaluate: $\int_1^9 \dfrac{x+1}{\sqrt{x}}\,dx$Free
  2. Q7Evaluate: $\int_2^3 \dfrac{1}{x^2+5x+6}\,dx$Free
  3. Q8Evaluate: $\int_0^{\pi/4} \cot^2 x\,dx$Free
  4. Q9Evaluate: $\int_{-\pi/4}^{\pi/4} \dfrac{1}{1-\sin x}\,dx$Preview
  5. Q10Evaluate: $\int_3^5 \dfrac{1}{\sqrt{2x+3}-\sqrt{2x-3}}\,dx$Preview
  6. Q11Evaluate: $\int_0^1 \dfrac{x^2-2}{x^2+1}\,dx$Preview
  7. Q12Evaluate: $\int_0^{\pi/4} \sin 4x \sin 3x\,dx$Preview
  8. Q13Evaluate: $\int_0^{\pi/4} \sqrt{1+\sin 2x}\,dx$Preview
  9. Q14Evaluate: $\int_0^{\pi/4} \sin^4 x\,dx$Preview
  10. Q15Evaluate: $\int_{-4}^{2} \dfrac{1}{x^2+4x+13}\,dx$Preview
  11. Q16Evaluate: $\int_0^4 \dfrac{1}{\sqrt{4x-x^2}}\,dx$Preview
  12. Q17Evaluate: $\int_0^1 \dfrac{1}{\sqrt{3+2x-x^2}}\,dx$Preview
  13. Q18Evaluate: $\int_0^{\pi/2} x\sin x\,dx$Preview
  14. Q19Evaluate: $\int_0^1 x\tan^{-1}x\,dx$Preview
  15. Q20Evaluate: $\int_0^\infty x e^{-x}\,dx$Preview
  16. Q21Evaluate: $\int_0^{1/\sqrt2} \dfrac{\sin^{-1}x}{(1-x^2)^{3/2}}\,dx$Preview
  17. Q22Evaluate: $\int_0^{\pi/4} \dfrac{\sec^2 x}{3\tan^2x+4\tan x+1}\,dx$Preview
  18. Q23Evaluate: $\int_0^{\pi/4} \dfrac{\sin 2x}{\sin^4x+\cos^4x}\,dx$Preview
  19. Q24Evaluate: $\int_0^{\pi/2} \sqrt{\cos x}\,\sin^3 x\,dx$Preview
  20. Q25Evaluate: $\int_0^{\pi/2} \dfrac{1}{5+4\cos x}\,dx$Preview
  21. Q26Evaluate: $\int_0^{\pi/4} \dfrac{\cos x}{4-\sin^2 x}\,dx$Preview
  22. Q27Evaluate: $\int_0^{\pi/2} \dfrac{\cos x}{(1+\sin x)(2+\sin x)}\,dx$Preview
  23. Q28Evaluate: $\int_{-1}^1 \dfrac{1}{a^2e^x+b^2e^{-x}}\,dx$Preview
  24. Q29Evaluate: $\int_0^\pi \dfrac{1}{3+2\sin x+\cos x}\,dx$Preview
  25. Q30Evaluate: $\int_0^{\pi/4} \sec^4 x\,dx$Preview
  26. Q31Evaluate: $\int_0^1 \sqrt{\dfrac{1-x}{1+x}}\,dx$Preview
  27. Q32Evaluate: $\int_0^\pi \dfrac{\sin 3x\,(1+2\cos x)}{(1+\cos x)^2}\,dx$Preview
  28. Q33Evaluate: $\int_0^{\pi/2} \sin 2x \,\tan^{-1}(\sin x)\,dx$Preview
  29. Q34Evaluate: $\int_{1/\sqrt2}^1 \dfrac{e^{\cos^{-1}x}\,\sin^{-1}x}{\sqrt{1-x^2}}\,dx$Preview
  30. Q35Evaluate: $\int_2^3 \dfrac{\cos(\log x)}{x}\,dx$Preview
  31. Q36Evaluate: $\int_0^a \dfrac{1}{x+\sqrt{a^2-x^2}}\,dx$Preview
  32. Q37Evaluate: $\int_0^{\pi/2} \log(\tan x)\,dx$Preview
  33. Q38Evaluate: $\int_0^1 \log\!\left(\dfrac{1}{x}-1\right)dx$Preview
  34. Q39Evaluate: $\int_0^{\pi/2} \dfrac{\sin x-\cos x}{1+\sin x\cos x}\,dx$Preview
  35. Q40Evaluate: $\int_0^3 x^2(3-x)^{5/2}\,dx$Preview
  36. Q41Evaluate: $\int_{-3}^3 \dfrac{x^3}{9-x^2}\,dx$Preview
  37. Q42Evaluate: $\int_{-\pi/2}^{\pi/2} \log\!\left(\dfrac{2+\sin x}{2-\sin x}\right)dx$Preview
  38. Q43Evaluate: $\int_{-\pi/4}^{\pi/4} \dfrac{x+\pi/4}{2-\cos 2x}\,dx$Preview
  39. Q44Evaluate: $\int_{-\pi/4}^{\pi/4} x^3\sin^4x\,dx$Preview
  40. Q45Evaluate: $\int_0^1 \dfrac{\log(x+1)}{x^2+1}\,dx$Preview
  41. Q46Evaluate: $\int_{-1}^1 \dfrac{x^3+2}{\sqrt{x^2+4}}\,dx$Preview
  42. Q47Evaluate: $\int_{-a}^a \dfrac{x+x^3}{16-x^2}\,dx$Preview
  43. Q48Evaluate: $\int_0^1 t^2\sqrt{1-t}\,dt$Preview
  44. Q49Evaluate: $\int_0^\pi x\sin x\cos^2 x\,dx$Preview
  45. Q50Evaluate: $\int_0^1 \dfrac{\log x}{\sqrt{1-x^2}}\,dx$Preview
4.2.1

Properties of Definite Integrals

These eight properties let a definite integral be simplified — often to zero, or to twice a simpler integral, or to another definite integral that is easier to handle — purely by symmetry or by a chan…

4.2.2

Solved Examples Using the Properties

This block of fifteen fully solved examples shows the properties of Section 4.2.1 combined with ordinary integration techniques (rationalising a surd denominator, half-angle/product-to-sum trigonometr…

4.2.3

Reduction Formulae for Definite Integrals of sin^n x and cos^n x

For the special family of integrals and (products of the same power of sine or cosine over a quarter period), there is a standard shortcut called a reduction formula that gives the value directly from…

Sample & Board Papers

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

+Show 21 questions21 questions
  1. Q1Evaluate: $\displaystyle\int_0^{\pi} \dfrac{x}{a^2\cos^2 x + b^2\sin^2 x}\,dx$Preview
  2. Q2Prove that: $\displaystyle\int_0^{2a} f(x)\,dx = \int_0^{a} f(x)\,dx + \int_0^{a} f(2a - x)\,dx$Preview
  3. Q3Evaluate: $\displaystyle\int_{-a}^{a} \sqrt{\dfrac{a-x}{a+x}}\,dx$Preview
  4. Q4If $\displaystyle\int_0^k \dfrac{1}{2+8x^2}\,dx = \dfrac{\pi}{16}$, then the value of $k$ is ______. (a) $\dfrac{1}{2}$ (b) $\dfrac{1}{3}$ (…Preview
  5. Q5Show that: $\displaystyle\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$, if $f(x)$ is an even function. $= 0$, if $f(x)$ is an odd function.Preview
  6. Q6Evaluate: $\displaystyle\int_0^{\pi/2} \sin^2 x\,\mathrm{d}x$Preview
  7. Q7Show that: $\displaystyle\int_0^{\pi/4}\log(1+\tan x)\,\mathrm{d}x = \dfrac{\pi}{8}\log 2$Preview
  8. Q8If $\displaystyle\int_0^k 4x^3\,dx = 16$, then the value of k is ________. (a) 1 (b) 2 (c) 3 (d) 4Preview
  9. Q9Evaluate: $\displaystyle\int_0^{\pi/2} \cos^2 x\,dx$Preview
  10. Q10Prove that: $\displaystyle\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a-x)\,dx$Preview
  11. Q11Evaluate: $\displaystyle\int_0^{\pi/4} \sec^4 x\,dx$Preview
  12. Q12Show that: $\displaystyle\int_0^{\pi/4} \log(1+\tan x)\,dx = \dfrac{\pi}{8}\log 2$Preview
  13. Q13Evaluate: $\displaystyle\int_0^{\pi/2} \sqrt{1-\cos 4x}\,dx$Preview
  14. Q14Prove that: $\displaystyle\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a-x)\,dx$Preview
  15. Q15If $\displaystyle\int_{-\pi/4}^{\pi/4} x^3 \sin^4 x \, dx = k$ then $k =$ ____. (a) 1 (b) 2 (c) 4 (d) 0Preview
  16. Q16Evaluate: $\displaystyle\int_0^{\pi/2} \cos^2 x \, dx$Preview
  17. Q17Prove that: $\displaystyle\int_0^{2a} f(x)\,dx=\int_0^a f(x)\,dx+\int_0^a f(2a-x)\,dx$. Hence show that: $\displaystyle\int_0^{\pi} \sin x\,…Preview
  18. Q18Evaluate: $\displaystyle\int_{-\pi/4}^{\pi/4}\dfrac{1}{1-\sin x}\,dx$Preview
  19. Q19Prove that: $\displaystyle\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$. Hence evaluate: $\displaystyle\int_0^3 \dfrac{\sqrt x}{\sqrt x+\sqrt{3-x…Preview
  20. Q20$\displaystyle\int_1^2 \dfrac{1}{x^2}\cdot e^{1/x}\,dx=$ ____. (a) $\sqrt e + 1$ (b) $\sqrt e - 1$ (c) $\sqrt e(\sqrt e - 1)$ (d) $\dfrac{\s…Preview
  21. Q21Prove that: $\displaystyle\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$. Hence, find $\displaystyle\int_{\pi/6}^{\pi/3} \sin^2 x\,dx$Preview

More questions

+Show 33 questions33 questions
  1. Q51Choose the correct option from the given alternatives: $\int_2^3 \dfrac{dx}{x(x^3-1)} =$ (A) $\frac13\log\frac{208}{189}$ (B) $\frac13\log\f…Free
  2. Q52Choose the correct option from the given alternatives: $\int_0^{\pi/2} \dfrac{\sin^2 x}{(1+\cos x)^2}\,dx =$ (A) $\frac{4-\pi}{2}$ (B) $\fra…Free
  3. Q53Choose the correct option from the given alternatives: $\int_0^{\log 5} \dfrac{e^x\sqrt{e^x-1}}{e^x+3}\,dx =$ (A) $3+2\pi$ (B) $4-\pi$ (C) $…Free
  4. Q54Choose the correct option from the given alternatives: $\int_0^{\pi/2} \sin^6 x \cos^2 x\,dx =$ (A) $\frac{7\pi}{256}$ (B) $\frac{3\pi}{256}…Preview
  5. Q55Choose the correct option from the given alternatives: If $\int_0^1 \dfrac{dx}{\sqrt{1+x}-\sqrt{x}} = \dfrac{k}{3}$, then $k$ is equal to (A…Preview
  6. Q56Choose the correct option from the given alternatives: $\int_1^2 \dfrac{1}{x^2}e^{1/x}\,dx =$ (A) $\sqrt e+1$ (B) $\sqrt e-1$ (C) $\sqrt e(\…Preview
  7. Q57Choose the correct option from the given alternatives: If $\int_2^e \left[\dfrac{1}{\log x}-\dfrac{1}{(\log x)^2}\right]dx = a+\dfrac{b}{\lo…Preview
  8. Q58Choose the correct option from the given alternatives: Let $I_1=\int_e^{e^2}\dfrac{dx}{\log x}$ and $I_2=\int_1^2 \dfrac{e^x}{x}\,dx$, then…Preview
  9. Q59Choose the correct option from the given alternatives: $\int_0^9 \dfrac{\sqrt x}{\sqrt x+\sqrt{9-x}}\,dx =$ (A) $9$ (B) $\frac92$ (C) $0$ (D…Preview
  10. Q60Choose the correct option from the given alternatives: The value of $\int_{-\pi/4}^{\pi/4} \log\left(\dfrac{2+\sin\theta}{2-\sin\theta}\righ…Preview
  11. Q61Evaluate: $\int_0^{\pi/2} \dfrac{\cos x}{3\cos x+\sin x}\,dx$Preview
  12. Q62Evaluate: $\int_{\pi/4}^{\pi/2} \dfrac{\cos\theta}{\left[\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\right]^3}\,d\theta$Preview
  13. Q63Evaluate: $\int_0^1 \dfrac{1}{1+\sqrt x}\,dx$Preview
  14. Q64Evaluate: $\int_0^{\pi/4} \dfrac{\tan^3 x}{1+\cos 2x}\,dx$Preview
  15. Q65Evaluate: $\int_0^1 t^5\sqrt{1-t^2}\,dt$Preview
  16. Q66Evaluate: $\int_0^1 (\cos^{-1}x)^2\,dx$Preview
  17. Q67Evaluate: $\int_{-1}^1 \dfrac{1+x^3}{9-x^2}\,dx$Preview
  18. Q68Evaluate: $\int_0^\pi x\sin x\cos^4 x\,dx$Preview
  19. Q69Evaluate: $\int_0^\pi \dfrac{x}{1+\sin^2 x}\,dx$Preview
  20. Q70Evaluate: $\int_1^\infty \dfrac{1}{\sqrt x\,(1+x)}\,dx$Preview
  21. Q71Evaluate: $\int_0^1 \dfrac{1}{1+x^2}\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)dx$Preview
  22. Q72Evaluate: $\int_0^{\pi/2} \dfrac{1}{6-\cos x}\,dx$Preview
  23. Q73Evaluate: $\int_0^a \dfrac{1}{a^2+ax-x^2}\,dx$Preview
  24. Q74Evaluate: $\int_{\pi/5}^{3\pi/10} \dfrac{\sin x}{\sin x+\cos x}\,dx$Preview
  25. Q75Evaluate: $\int_0^1 \sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)dx$Preview
  26. Q76Evaluate: $\int_0^{\pi/4} \dfrac{\cos 2x}{1+\cos 2x+\sin 2x}\,dx$Preview
  27. Q77Evaluate: $\int_0^{\pi/2} (2\log\sin x - \log\sin 2x)\,dx$Preview
  28. Q78Evaluate: $\int_0^\pi (\sin^{-1}x+\cos^{-1}x)^3\sin^3 x\,dx$Preview
  29. Q79Evaluate: $\int_0^4 \left[\sqrt{x^2+2x+3}\right]^{-1}dx$Preview
  30. Q80Evaluate: $\int_{-2}^3 |x-2|\,dx$Preview
  31. Q81If $\int_0^a \sqrt{x}\,dx = 2a\displaystyle\int_0^{\pi/2}\sin^3x\,dx$, then find the value of $\int_a^{a+1} x\,dx$Preview
  32. Q82If $\int_0^k \dfrac{1}{2+8x^2}\,dx = \dfrac{\pi}{16}$, find $k$.Preview
  33. Q83If $f(x)=a+bx+cx^2$, show that $\int_0^1 f(x)\,dx = \dfrac16\left[f(0)+4f\!\left(\dfrac12\right)+f(1)\right]$.Preview