Skip to content
← Mathematics

Mathematics · Class 12 Science

Ch 10Indefinite Integration — Class 12 Mathematics, concept-first.

In differential calculus we learned how to find the derivative of a function. Indefinite integration asks the reverse question: given a function , can we find a function whose derivative is ? If , we call a primitive (or antiderivative) of , and we write

255

Q&A

13

Concepts

10m

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Integration by Substitution

The substitution (change-of-variable) method mirrors the chain rule of differentiation. If is a differentiable function, then because .

Most relevant Q&A

Start with this concept →

In previous exams

How often this chapter’s concepts have been examined — real appearance data, never estimated.

CA Foundation

2 concepts · 3 appearances

2025, 2024

MHT-CET

1 concept · 30 appearances

2026, 2025, 2024, 2023, 2022, 2021, 2020

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction to Indefinite Integration

In differential calculus we learned how to find the derivative of a function. Indefinite integration asks the reverse question: given a function , can we find a function whose derivative is ? If , we…

3.1.1

Elementary Integration Formulae

Every differentiation formula can be read in reverse to produce an integration formula. If , then automatically .

3.1.2

Properties of Indefinite Integrals and Direct Applications

Three simple but powerful properties let us break a complicated integrand apart and integrate it piece by piece.

3.1.3

Reducing an Integral to Standard Form by Simplification

Many trigonometric or surd integrands look intractable at first glance, yet collapse to an elementary formula once a suitable identity or algebraic trick is used.

3.2

Methods of Integration — Overview

So far every integral has been reduced to a standard form purely by algebraic or trigonometric rewriting.

3.2.1

Integration by Substitution

Theorem. If is a differentiable function of , then .

3.2.2

Integrals of tan x, cot x, sec x and cosec x

Corollary III of section 3.2.1 () immediately settles the integrals of , , and .

3.2.3

Some Special Integrals

The seven special integrals listed above are each proved by a trigonometric substitution, chosen according to which quadratic surd or quadratic denominator is present (see the substitution table).

3.2.4

Integrals of the Form $\int\dfrac{dx}{ax^2+bx+c}$ and $\int\dfrac{dx}{\sqrt{ax^2+bx+c}}$

To evaluate or , follow five steps: (1) write as and pull (or ) outside the integral; (2) complete the square on by adding and subtracting ; (3) write the result as a sum or difference of two squares,…

3.2.5

Integrals of the Form $\int\dfrac{dx}{a\sin^2x+b\cos^2x+c}$

To evaluate : (1) divide numerator and denominator by ; (2) replace by in the denominator; (3) substitute , reducing the integral to the quadratic-denominator form handled in 3.2.4; (4) evaluate using…

3.2.6

Integrals of the Form $\int\dfrac{dx}{a\sin x+b\cos x+c}$

To evaluate , substitute . Then , so (using ); and, from the double-angle formulae applied to , and . Substituting turns the whole integral into a rational function of with a quadratic denominator — t…

3.2.7

Integrals of the Form $\int\dfrac{px+q}{ax^2+bx+c}\,dx$ and $\int\dfrac{px+q}{\sqrt{ax^2+bx+c}}\,dx$

When the numerator is linear, write for constants found by comparing coefficients. This splits into — the first piece is a Corollary-III log (put ), the second is the 3.2.4 type.

3.3

Integration by Parts — Overview

Integration by parts undoes the product rule for derivatives. It is used whenever the integrand is naturally a product of two functions of different kinds — a polynomial times a trigonometric function…

3.3.1

The Integration-by-Parts Theorem and the LIATE Rule

Theorem. If are differentiable functions of , then .

3.3.2

Integrating $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$, $\sqrt{x^2-a^2}$ and $(px+q)\sqrt{ax^2+bx+c}$

. Treat as a product with : by parts with , . Write : the remaining integral splits into . So , giving the standard result .

3.3.3

Integral of the Form $\int e^x[f(x)+f'(x)]\,dx$

Claim: .

3.4

Integration by Partial Fractions

A rational function () is called proper if , and improper otherwise; an improper fraction is first written as by polynomial division, where the remainder term is proper.

3.5

Something Interesting: A Generalised Rule for Repeated Integration by Parts

Ordinary integration by parts is , where follow the LIATE order. When is a polynomial, applying this rule over and over eventually terminates, because some high-enough derivative of a polynomial is ex…

Sample & Board Papers

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

+Show 35 questions35 questions
  1. Q1Evaluate: $\displaystyle\int \dfrac{\sin x}{\sqrt{36 - \cos^2 x}}\,dx$Preview
  2. Q2Prove that: $\displaystyle\int \sqrt{a^2 - x^2}\,dx = \dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\left(\dfrac{x}{a}\right) + c$Preview
  3. Q3Evaluate: $\displaystyle\int \dfrac{1 + \log x}{x(2 + \log x)(3 + \log x)}\,dx$Preview
  4. Q4Evaluate: $\displaystyle\int \dfrac{\sin\sqrt{x}}{\sqrt{x}}\,dx$Preview
  5. Q5If u and v are two functions of $x$, then prove that: $\displaystyle\int u\,v\,dx = u\int v\,dx - \int\left[\dfrac{du}{dx}\int v\,dx\right]d…Preview
  6. Q6Evaluate: $\displaystyle\int \dfrac{d\theta}{\sin\theta + \sin 2\theta}$Preview
  7. Q7Evaluate: $\displaystyle\int e^x\left[\dfrac{\cos x - \sin x}{\sin^2 x}\right]dx$.Preview
  8. Q8Evaluate: $\displaystyle\int \dfrac{1}{3+2\sin x+\cos x}\,dx$.Preview
  9. Q9Prove that: $\displaystyle\int \dfrac{1}{a^2-x^2}\,dx = \dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right| + c$.Preview
  10. Q10$\displaystyle\int \dfrac{\mathrm{d}x}{9x^2+1} =$________. (a) $\dfrac{1}{3}\tan^{-1}(2x)+c$ (b) $\dfrac{1}{3}\tan^{-1}x+c$ (c) $\dfrac{1}{3…Preview
  11. Q11Evaluate: $\displaystyle\int \dfrac{\mathrm{e}^x(1+x)}{\cos^2(x\mathrm{e}^x)}\,\mathrm{d}x$Preview
  12. Q12Prove that: $\displaystyle\int \dfrac{\mathrm{d}x}{\sqrt{x^2+a^2}} = \log\left|x+\sqrt{x^2+a^2}\right|+c$Preview
  13. Q13Evaluate: $\displaystyle\int \dfrac{dx}{x^2+4x+8}$Preview
  14. Q14Evaluate: $\displaystyle\int \dfrac{x^2 \cdot \tan^{-1}(x^3)}{1+x^6}\,dx$Preview
  15. Q15Prove that: $\displaystyle\int \sqrt{x^2+a^2}\,dx = \dfrac{x}{2}\sqrt{x^2+a^2}+\dfrac{a^2}{2}\log\left|x+\sqrt{x^2+a^2}\right|+c$Preview
  16. Q16If $\displaystyle\int \dfrac{dx}{4x^2-1} = A\log\left(\dfrac{2x-1}{2x+1}\right)+c$, then $A = $ ________. (a) 1 (b) $\dfrac{1}{2}$ (c) $\dfr…Preview
  17. Q17If $f'(x) = x^{-1}$, then find $f(x)$Preview
  18. Q18Evaluate: $\displaystyle\int \dfrac{dx}{2+\cos x - \sin x}$Preview
  19. Q19If $u$ and $v$ are differentiable functions of $x$, then prove that: $\displaystyle\int uv\,dx = u\int v\,dx - \int\left[\dfrac{du}{dx}\int…Preview
  20. Q20$\displaystyle\int \cos^3 x\,dx = $ ________. (a) $\dfrac{1}{12}\sin 3x + \dfrac{3}{4}\sin x + c$ (b) $\dfrac{1}{12}\sin 3x + \dfrac{1}{4}\s…Preview
  21. Q21Write $\displaystyle\int \cot x\,dx$.Preview
  22. Q22Evaluate $\displaystyle\int x\tan^{-1}x\,dx$Preview
  23. Q23Evaluate: $\displaystyle\int \dfrac{2x^2-3}{(x^2-5)(x^2+4)}\,dx$Preview
  24. Q24Evaluate: $\displaystyle\int \dfrac{1}{x^2+25}\,dx$Preview
  25. Q25Evaluate: $\displaystyle\int \log x \, dx$.Preview
  26. Q26Prove that: $\displaystyle\int \dfrac{1}{a^2-x^2}\,dx=\dfrac{1}{2a}\log\left(\dfrac{a+x}{a-x}\right)+c$.Preview
  27. Q27Evaluate: $\displaystyle\int \dfrac{5e^x}{(e^x+1)(e^{2x}+9)}\,dx$Preview
  28. Q28The value of $\displaystyle\int x^x(1+\log x)\,dx$ is equal to ____. (a) $\dfrac{1}{2}(1+\log x)^2+c$ (b) $x^{2x}+c$ (c) $x^x\cdot\log x + c…Preview
  29. Q29Evaluate: $\displaystyle\int x^9\cdot\sec^2(x^{10})\,dx$Preview
  30. Q30Evaluate: $\displaystyle\int \dfrac{1}{25-9x^2}\,dx$Preview
  31. Q31Evaluate: $\displaystyle\int e^{\sin^{-1}x}\left(\dfrac{x+\sqrt{1-x^2}}{\sqrt{1-x^2}}\right)dx$Preview
  32. Q32Evaluate: $\displaystyle\int \dfrac{2x}{1+x^2}\,dx$Preview
  33. Q33Evaluate: $\displaystyle\int \sqrt{1+\sin 2x}\cdot dx$Preview
  34. Q34Evaluate: $\displaystyle\int \dfrac{\sin x}{\sin 3x}\,dx$Preview
  35. Q35Evaluate: $\displaystyle\int \dfrac{3x^2+4x-5}{(x^2-1)(x+2)}\,dx$Preview

More questions

220 Q
+Show 49 questions49 questions
  1. Q172Choose the correct option: $\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}dx =$ (A) $\frac12\sqrt{x+1}+c$ (B) $\frac23(x+1)^{3/2}+c$ (C)…Free
  2. Q173If $\int \frac{1}{x+x^5}dx = f(x)+c$, then $\int \frac{x^4}{x+x^5}dx =$ (A) $\log x - f(x)+c$ (B) $f(x)+\log x+c$ (C) $f(x)-\log x+c$ (D) $\…Free
  3. Q174Choose the correct option: $\int \frac{\log(3x)}{x\log(9x)}dx =$ (A) $\log(3x)-\log(9x)+c$ (B) $\log(x)-(\log 3)\log(\log 9x)+c$ (C) $\log 9…Free
  4. Q175Choose the correct option: $\int \frac{\sin^m x}{\cos^{m+2}x}dx =$ (A) $\frac{\tan^{m+1}x}{m+1}+c$ (B) $(m+2)\tan^{m+1}x+c$ (C) $\frac{\tan^…Preview
  5. Q176Choose the correct option: $\int \tan(\sin^{-1}x)\,dx =$ (A) $(1-x^2)^{-1/2}+c$ (B) $(1-x^2)^{1/2}+c$ (C) $\frac{\tan^m x}{\sqrt{1-x^2}}+c$…Preview
  6. Q177Choose the correct option: $\int \frac{x-\sin x}{1-\cos x}dx =$ (A) $x\cot\frac{x}{2}+c$ (B) $-x\cot\frac{x}{2}+c$ (C) $\cot\frac{x}{2}+c$ (…Preview
  7. Q178If $f(x)=\frac{\sin^{-1}x}{\sqrt{1-x^2}}$, $g(x)=e^{\sin^{-1}x}$, then $\int f(x)g(x)dx=$ (A) $e^{\sin^{-1}x}(\sin^{-1}x-1)+c$ (B) $e^{\sin^…Preview
  8. Q179If $\int \tan^3 x\sec^3 x\,dx = \frac{1}{m}\sec^m x - \frac{1}{n}\sec^n x+c$, then $(m,n)=$ (A) $(5,3)$ (B) $(3,5)$ (C) $\left(\frac15,\frac…Preview
  9. Q180Choose the correct option: $\int \frac{1}{\cos x-\cos^2 x}dx =$ (A) $\log(\csc x-\cot x)+\tan\frac{x}{2}+c$ (B) $\sin 2x-\cos x+c$ (C) $\log…Preview
  10. Q181Choose the correct option: $\int \frac{\sqrt{\cot x}}{\sin x\cos x}dx =$ (A) $2\sqrt{\cot x}+c$ (B) $-2\sqrt{\cot x}+c$ (C) $\frac12\sqrt{\c…Preview
  11. Q182Choose the correct option: $\int \frac{e^x(x-1)}{x^2}dx =$ (A) $\frac{e^x}{x}+c$ (B) $\frac{e^x}{x^2}+c$ (C) $\left(x-\frac1x\right)e^x+c$ (…Preview
  12. Q183Choose the correct option: $\int \sin(\log x)\,dx =$ (A) $\frac{x}{2}[\sin(\log x)-\cos(\log x)]+c$ (B) $\frac{x}{2}[\sin(\log x)+\cos(\log…Preview
  13. Q184Choose the correct option: $\int x^x(1+\log x)\,dx =$ (A) $\frac12(1+\log x)^2+c$ (B) $x^{2x}+c$ (C) $x^x\log x+c$ (D) $x^x+c$Preview
  14. Q185Choose the correct option: $\int \cos^{-3/7}x\sin^{-11/7}x\,dx =$ (A) $\log\left(\sin^{-4/7}x\right)+c$ (B) $\frac47\tan^{4/7}x+c$ (C) $-\fr…Preview
  15. Q186Choose the correct option: $2\int \frac{\cos^2 x-\sin^2 x}{\cos^2 x+\sin^2 x}dx =$ (A) $\sin 2x+c$ (B) $\cos 2x+c$ (C) $\tan 2x+c$ (D) $2\si…Preview
  16. Q187Choose the correct option: $\int \frac{dx}{\cos x\sqrt{\sin^2 x-\cos^2 x}} =$ (A) $\log(\tan x-\sqrt{\tan^2 x-1})+c$ (B) $\sin^{-1}(\tan x)+…Preview
  17. Q188Choose the correct option: $\int \frac{\log x}{(\log ex)^2}dx =$ (A) $\frac{x}{1+\log x}+c$ (B) $x(1+\log x)+c$ (C) $\frac{1}{1+\log x}+c$ (…Preview
  18. Q189Choose the correct option: $\int [\sin(\log x)+\cos(\log x)]\,dx =$ (A) $x\cos(\log x)+c$ (B) $\sin(\log x)+c$ (C) $\cos(\log x)+c$ (D) $x\s…Preview
  19. Q190Choose the correct option: $\int \frac{\cos 2x-1}{\cos 2x+1}dx =$ (A) $\tan x-x+c$ (B) $x+\tan x+c$ (C) $x-\tan x+c$ (D) $-x-\cot x+c$Preview
  20. Q191Choose the correct option: $\int \frac{e^{2x}+e^{-2x}}{e^x}dx =$ (A) $e^x-\frac{1}{3e^{3x}}+c$ (B) $e^x+\frac{1}{3e^{3x}}+c$ (C) $e^{-x}+\fr…Preview
  21. Q192Integrate: $\int (x-2)^2\sqrt{x}\,dx$Preview
  22. Q193Integrate: $\int \frac{x^7}{x+1}\,dx$Preview
  23. Q194Integrate: $\int (6x+5)^{3/2}\,dx$Preview
  24. Q195Integrate with respect to $t$: $\int \frac{t^3}{(t+1)^2}\,dt$Preview
  25. Q196Integrate: $\int \frac{3-2\sin x}{\cos^2 x}\,dx$Preview
  26. Q197Integrate with respect to $\theta$: $\int \frac{\sin^6\theta+\cos^6\theta}{\sin^2\theta\cos^2\theta}\,d\theta$Preview
  27. Q198Integrate: $\int \cos 3x\cos 2x\cos x\,dx$Preview
  28. Q199Integrate: $\int \frac{\cos 7x-\cos 8x}{1+2\cos 5x}\,dx$Preview
  29. Q200Integrate: $\int \cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)\,dx$Preview
  30. Q201Integrate: $\int \frac{(1+\log x)^3}{x}\,dx$Preview
  31. Q202Integrate: $\int \cot^{-1}(1-x+x^2)\,dx$Preview
  32. Q203Integrate: $\int \frac{1}{x\sin^2(\log x)}\,dx$Preview
  33. Q204Integrate: $\int \sqrt{x}\sec(x^{3/2})\tan(x^{3/2})\,dx$Preview
  34. Q205Integrate: $\int \left[\log(1+\cos x)-x\tan\frac{x}{2}\right]dx$Preview
  35. Q206Integrate: $\int \frac{x^2}{\sqrt{1-x^6}}\,dx$Preview
  36. Q207Integrate: $\int \frac{1}{(1-\cos 4x)(3-\cot 2x)}\,dx$Preview
  37. Q208Integrate: $\int \left[\log(\log x)+\frac{1}{(\log x)^2}\right]dx$Preview
  38. Q209Integrate: $\int \frac{1}{2\cos x+3\sin x}\,dx$Preview
  39. Q210Integrate: $\int \frac{1}{x^3\sqrt{x^2-1}}\,dx$Preview
  40. Q211Integrate: $\int \frac{3x+1}{\sqrt{-2x^2+x+3}}\,dx$Preview
  41. Q212Integrate: $\int \log(x^2+1)\,dx$Preview
  42. Q213Integrate: $\int e^{2x}\sin x\cos x\,dx$Preview
  43. Q214Integrate: $\int \frac{x^2}{(x-1)(3x-1)(3x-2)}\,dx$Preview
  44. Q215Integrate: $\int \frac{1}{\sin x+\sin 2x}\,dx$Preview
  45. Q216Integrate: $\int \sec^2 x\sqrt{7+2\tan x-\tan^2 x}\,dx$Preview
  46. Q217Integrate: $\int \frac{x+5}{x^3+3x^2-x-3}\,dx$Preview
  47. Q218Integrate: $\int \frac{1}{x(x^5+1)}\,dx$Preview
  48. Q219Integrate: $\int \frac{\sqrt{\tan x}}{\sin x\cos x}\,dx$Preview
  49. Q220Integrate: $\int \sec^4 x\csc^2 x\,dx$Preview
+Show 26 questions26 questions
  1. Q1Evaluate: $\int (x^3+x^2-x+1)\,dx$Free
  2. Q2Evaluate: $\int x^2\left(1-\dfrac{2}{x}\right)^2 dx$Free
  3. Q3Evaluate: $\int \left(3\sec^2 x - \dfrac{4}{x} + \dfrac{1}{x\sqrt{x}} - 7\right) dx$Free
  4. Q4Evaluate: $\int \left(\dfrac{2x^3-5x+3}{x} + \dfrac{4}{x^5}\right) dx$Preview
  5. Q5Evaluate: $\int \dfrac{3x^3-2x+5}{x\sqrt{x}}\,dx$Preview
  6. Q6Evaluate: $\int \tan^2 x\,dx$Preview
  7. Q7Evaluate: $\int \dfrac{\sin 2x}{\cos x}\,dx$Preview
  8. Q8Evaluate: $\int \dfrac{\sin x}{\cos^2 x}\,dx$Preview
  9. Q9Evaluate: $\int \dfrac{\cos 2x}{\sin^2 x}\,dx$Preview
  10. Q10Evaluate: $\int \dfrac{\cos 2x}{\sin^2 x\cos^2 x}\,dx$Preview
  11. Q11Evaluate: $\int \dfrac{\sin x}{1+\sin x}\,dx$Preview
  12. Q12Evaluate: $\int \dfrac{\tan x}{\sec x+\tan x}\,dx$Preview
  13. Q13Evaluate: $\int \sqrt{1+\sin 2x}\,dx$Preview
  14. Q14Evaluate: $\int \sqrt{1-\cos 2x}\,dx$Preview
  15. Q15Evaluate: $\int \sin 4x\cos 3x\,dx$Preview
  16. Q16Evaluate: $\int \dfrac{x}{x+2}\,dx$Preview
  17. Q17Evaluate: $\int \dfrac{4x+3}{2x+1}\,dx$Preview
  18. Q18Evaluate: $\int \dfrac{5x+2}{3x-4}\,dx$Preview
  19. Q19Evaluate: $\int \dfrac{x-2}{\sqrt{x+5}}\,dx$Preview
  20. Q20Evaluate: $\int \dfrac{2x-7}{\sqrt{4x-1}}\,dx$Preview
  21. Q21Evaluate: $\int \dfrac{\sin 4x}{\cos 2x}\,dx$Preview
  22. Q22Evaluate: $\int \sqrt{1+\sin 5x}\,dx$Preview
  23. Q23Evaluate: $\int \cos^2 x\,dx$Preview
  24. Q24Evaluate: $\int \dfrac{2}{\sqrt{x}-\sqrt{x+3}}\,dx$Preview
  25. Q25Evaluate: $\int \dfrac{3}{\sqrt{7x-2}-\sqrt{7x-5}}\,dx$Preview
  26. Q26Given $f'(x) = x - \dfrac{3}{x^3}$ and $f(1) = \dfrac{11}{2}$, find $f(x)$.Preview
+Show 42 questions42 questions
  1. Q27Integrate: $\int \dfrac{(\log x)^n}{x}\,dx$Free
  2. Q28Integrate: $\int \dfrac{(\sin^{-1}x)^{3/2}}{\sqrt{1-x^2}}\,dx$Free
  3. Q29Integrate: $\int \dfrac{1+x}{x\sin(x+\log x)}\,dx$Free
  4. Q30Integrate: $\int \dfrac{x\sec^2(x^2)}{\sqrt{\tan^3(x^2)}}\,dx$Preview
  5. Q31Integrate: $\int \dfrac{e^{3x}}{e^{3x}+1}\,dx$Preview
  6. Q32Integrate: $\int \dfrac{x^2+2}{x^2+1}\cdot a^{x+\tan^{-1}x}\,dx$Preview
  7. Q33Integrate: $\int \dfrac{e^x\log(\sin e^x)}{\tan e^x}\,dx$Preview
  8. Q34Integrate: $\int \dfrac{e^{2x}+1}{e^{2x}-1}\,dx$Preview
  9. Q35Integrate: $\int \sin^4 x\cos^3 x\,dx$Preview
  10. Q36Integrate: $\int \dfrac{1}{4x+5x^{-11}}\,dx$Preview
  11. Q37Integrate: $\int x^9\sec^2(x^{10})\,dx$Preview
  12. Q38Integrate: $\int e^{3\log x}\cdot\dfrac{1}{x^4+1}\,dx$Preview
  13. Q39Integrate: $\int \dfrac{\sqrt{\tan x}}{\sin x\cos x}\,dx$Preview
  14. Q40Integrate: $\int \dfrac{(x-1)^2}{(x^2+1)^2}\,dx$Preview
  15. Q41Integrate: $\int \dfrac{2\sin x\cos x}{3\cos^2 x+4\sin^2 x}\,dx$Preview
  16. Q42Integrate: $\int \dfrac{1}{\sqrt{x}+\sqrt[3]{x}}\,dx$Preview
  17. Q43Integrate: $\int \dfrac{10x^9+10^x\log 10}{10^x+x^{10}}\,dx$Preview
  18. Q44Integrate: $\int \dfrac{x^{n-1}}{\sqrt{1+4x^n}}\,dx$Preview
  19. Q45Integrate: $\int (2x+1)\sqrt{x+2}\,dx$Preview
  20. Q46Integrate: $\int x^5\sqrt{a^2+x^2}\,dx$Preview
  21. Q47Integrate: $\int \dfrac{5-3x}{\sqrt{2-3x}}\,dx$Preview
  22. Q48Integrate: $\int \dfrac{7+4x+5x^2}{(2x+3)^{3/2}}\,dx$Preview
  23. Q49Integrate: $\int \dfrac{x^2}{\sqrt{9-x^6}}\,dx$Preview
  24. Q50Integrate: $\int \dfrac{1}{x(x^3-1)}\,dx$Preview
  25. Q51Integrate: $\int \dfrac{1}{x\log x\log(\log x)}\,dx$Preview
  26. Q52Integrate: $\int \dfrac{\cos 3x-\cos 4x}{\sin 3x+\sin 4x}\,dx$Preview
  27. Q53Integrate: $\int \dfrac{\cos x}{\sin(x-a)}\,dx$Preview
  28. Q54Integrate: $\int \dfrac{\sin(x-a)}{\cos(x+b)}\,dx$Preview
  29. Q55Integrate: $\int \dfrac{1}{\sin x\cos x+2\cos^2 x}\,dx$Preview
  30. Q56Integrate: $\int \dfrac{\sin x+2\cos x}{3\sin x+4\cos x}\,dx$Preview
  31. Q57Integrate: $\int \dfrac{1}{2+3\tan x}\,dx$Preview
  32. Q58Integrate: $\int \dfrac{4e^x-25}{2e^x-5}\,dx$Preview
  33. Q59Integrate: $\int \dfrac{20+12e^x}{3e^x+4}\,dx$Preview
  34. Q60Integrate: $\int \dfrac{3e^{2x}+5}{4e^{2x}-5}\,dx$Preview
  35. Q61Integrate: $\int \cos^8 x\cot x\,dx$Preview
  36. Q62Integrate: $\int \tan^5 x\,dx$Preview
  37. Q63Integrate: $\int \cos^7 x\,dx$Preview
  38. Q64Integrate: $\int \tan 3x\tan 2x\tan x\,dx$Preview
  39. Q65Integrate: $\int \sin^5 x\cos^8 x\,dx$Preview
  40. Q66Integrate: $\int 3\cos^2 x\sin 2x\,dx$Preview
  41. Q67Integrate: $\int \dfrac{\sin 6x}{\sin 10x\sin 4x}\,dx$Preview
  42. Q68Integrate: $\int \dfrac{\sin x\cos^3 x}{1+\cos 2x}\,dx$Preview
+Show 29 questions29 questions
  1. Q69Evaluate: $\int \frac{1}{4x^2-3}\,dx$Free
  2. Q70Evaluate: $\int \frac{1}{25-9x^2}\,dx$Free
  3. Q71Evaluate: $\int \frac{1}{7+2x^2}\,dx$Free
  4. Q72Evaluate: $\int \frac{1}{\sqrt{3x^2+8}}\,dx$Preview
  5. Q73Evaluate: $\int \frac{1}{\sqrt{11-4x^2}}\,dx$Preview
  6. Q74Evaluate: $\int \frac{1}{\sqrt{2x^2-5}}\,dx$Preview
  7. Q75Evaluate: $\int \sqrt{\dfrac{9+x}{9-x}}\,dx$Preview
  8. Q76Evaluate: $\int \sqrt{\dfrac{2+x}{2-x}}\,dx$Preview
  9. Q77Evaluate: $\int \sqrt{\dfrac{10+x}{10-x}}\,dx$Preview
  10. Q78Evaluate: $\int \frac{1}{x^2+8x+12}\,dx$Preview
  11. Q79Evaluate: $\int \frac{1}{1+x-x^2}\,dx$Preview
  12. Q80Evaluate: $\int \frac{1}{4x^2-20x+17}\,dx$Preview
  13. Q81Evaluate: $\int \frac{1}{5-4x-3x^2}\,dx$Preview
  14. Q82Evaluate: $\int \frac{1}{\sqrt{3x^2+5x+7}}\,dx$Preview
  15. Q83Evaluate: $\int \frac{1}{\sqrt{x^2+8x-20}}\,dx$Preview
  16. Q84Evaluate: $\int \frac{1}{\sqrt{8-3x+2x^2}}\,dx$Preview
  17. Q85Evaluate: $\int \frac{1}{\sqrt{(x-3)(x+2)}}\,dx$Preview
  18. Q86Evaluate: $\int \frac{1}{4+3\cos^2 x}\,dx$Preview
  19. Q87Evaluate: $\int \frac{1}{\cos 2x+3\sin^2 x}\,dx$Preview
  20. Q88Evaluate: $\int \frac{\sin x}{\sin 3x}\,dx$Preview
  21. Q89Integrate: $\int \frac{1}{3+2\sin x}\,dx$Preview
  22. Q90Integrate: $\int \frac{1}{4-5\cos x}\,dx$Preview
  23. Q91Integrate: $\int \frac{1}{2+\cos x-\sin x}\,dx$Preview
  24. Q92Integrate: $\int \frac{1}{3+2\sin x-\cos x}\,dx$Preview
  25. Q93Integrate: $\int \frac{1}{3-2\cos 2x}\,dx$Preview
  26. Q94Integrate: $\int \frac{1}{2\sin 2x-3}\,dx$Preview
  27. Q95Integrate: $\int \frac{1}{3+2\sin 2x+4\cos 2x}\,dx$Preview
  28. Q96Integrate: $\int \frac{1}{\cos x-\sin x}\,dx$Preview
  29. Q97Integrate: $\int \frac{1}{\cos x-\sqrt3\sin x}\,dx$Preview
+Show 9 questions9 questions
  1. Q98Evaluate: $\int \frac{3x+4}{x^2+6x+5}\,dx$Free
  2. Q99Evaluate: $\int \frac{2x+1}{x^2+4x-5}\,dx$Free
  3. Q100Evaluate: $\int \frac{2x+3}{2x^2+3x-1}\,dx$Free
  4. Q101Evaluate: $\int \frac{3x+4}{\sqrt{2x^2+2x+1}}\,dx$Preview
  5. Q102Evaluate: $\int \frac{7x+3}{\sqrt{3+2x-x^2}}\,dx$Preview
  6. Q103Evaluate: $\int \frac{x-7}{x-9}\,dx$Preview
  7. Q104Evaluate: $\int \sqrt{\dfrac{9-x}{x}}\,dx$Preview
  8. Q105Evaluate: $\int \frac{3\cos x}{4\sin^2 x+4\sin x-1}\,dx$Preview
  9. Q106Evaluate: $\int \frac{e^{3x}-e^{2x}}{e^x+1}\,dx$Preview
+Show 42 questions42 questions
  1. Q107Evaluate: $\int x^2\log x\,dx$Free
  2. Q108Evaluate: $\int x^2\sin 3x\,dx$Free
  3. Q109Evaluate: $\int x\tan^{-1}x\,dx$Free
  4. Q110Evaluate: $\int x^2\tan^{-1}x\,dx$Preview
  5. Q111Evaluate: $\int x^3\tan^{-1}x\,dx$Preview
  6. Q112Evaluate: $\int (\log x)^2\,dx$Preview
  7. Q113Evaluate: $\int \sec^3 x\,dx$Preview
  8. Q114Evaluate: $\int x\sin^2 x\,dx$Preview
  9. Q115Evaluate: $\int x^3\log x\,dx$Preview
  10. Q116Evaluate: $\int e^{2x}\cos 3x\,dx$Preview
  11. Q117Evaluate: $\int x\sin^{-1}x\,dx$Preview
  12. Q118Evaluate: $\int x^2\cos^{-1}x\,dx$Preview
  13. Q119Evaluate: $\int \frac{\log(\log x)}{x}\,dx$Preview
  14. Q120Evaluate: $\int \frac{t\sin^{-1}t}{\sqrt{1-t^2}}\,dt$Preview
  15. Q121Evaluate: $\int \cos\sqrt{x}\,dx$Preview
  16. Q122Evaluate: $\int \sin\theta\log(\cos\theta)\,d\theta$Preview
  17. Q123Evaluate: $\int x\cos^3 x\,dx$Preview
  18. Q124Evaluate: $\int \frac{\sin\left((\log x)^2\right)}{x}\cdot\log x\,dx$Preview
  19. Q125Evaluate: $\int \frac{\log x}{x}\,dx$Preview
  20. Q126Evaluate: $\int x\sin 2x\cos 5x\,dx$Preview
  21. Q127Evaluate: $\int \cos(\sqrt[3]{x})\,dx$Preview
  22. Q128Evaluate: $\int e^{2x}\sin 3x\,dx$Preview
  23. Q129Evaluate: $\int e^{-x}\cos 2x\,dx$Preview
  24. Q130Evaluate: $\int \sin(\log x)\,dx$Preview
  25. Q131Evaluate: $\int \sqrt{5x^2+3}\,dx$Preview
  26. Q132Evaluate: $\int x^2\sqrt{a^2-x^6}\,dx$Preview
  27. Q133Evaluate: $\int \sqrt{(x-3)(7-x)}\,dx$Preview
  28. Q134Evaluate: $\int \sqrt{4x(4x+4)}\,dx$Preview
  29. Q135Evaluate: $\int (x+1)\sqrt{2x^2+3}\,dx$Preview
  30. Q136Evaluate: $\int x\sqrt{5-4x-x^2}\,dx$Preview
  31. Q137Evaluate: $\int \sec^2 x\sqrt{\tan^2 x+\tan x-7}\,dx$Preview
  32. Q138Evaluate: $\int \sqrt{x^2+2x+5}\,dx$Preview
  33. Q139Evaluate: $\int \sqrt{2x^2+3x+4}\,dx$Preview
  34. Q140Evaluate: $\int (2+\cot x-\csc^2 x)e^x\,dx$Preview
  35. Q141Evaluate: $\int \frac{1+\sin x}{1+\cos x}\cdot e^x\,dx$Preview
  36. Q142Evaluate: $\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right)dx$Preview
  37. Q143Evaluate: $\int \frac{x}{(x+1)^2}\cdot e^x\,dx$Preview
  38. Q144Evaluate: $\int \frac{e^x}{x}\left[x(\log x)^2+2\log x\right]dx$Preview
  39. Q145Evaluate: $\int e^{5x}\cdot\frac{5x\log x+1}{x}\,dx$Preview
  40. Q146Evaluate: $\int e^{\sin^{-1}x}\cdot\frac{x+\sqrt{1-x^2}}{\sqrt{1-x^2}}\,dx$Preview
  41. Q147Evaluate: $\int \frac{\log(1+x)}{1+x}\,dx$Preview
  42. Q148Evaluate: $\int \csc(\log x)\left[1-\cot(\log x)\right]dx$Preview
+Show 23 questions23 questions
  1. Q149Evaluate: $\int \frac{x^2+2}{(x-1)(x+2)(x+3)}\,dx$Free
  2. Q150Evaluate: $\int \frac{x^2}{(x^2+1)(x^2-2)(x^2+3)}\,dx$Free
  3. Q151Evaluate: $\int \frac{12x+3}{6x^2+13x-63}\,dx$Free
  4. Q152Evaluate: $\int \frac{2x}{4-3x-x^2}\,dx$Preview
  5. Q153Evaluate: $\int \frac{x^2+x-1}{x^2+x-6}\,dx$Preview
  6. Q154Evaluate: $\int \frac{6x^3+5x^2-7}{3x^2-2x-1}\,dx$Preview
  7. Q155Evaluate: $\int \frac{12x^2-2x-9}{(4x^2-1)(x+3)}\,dx$Preview
  8. Q156Evaluate: $\int \frac{1}{x(x^5+1)}\,dx$Preview
  9. Q157Evaluate: $\int \frac{2x^2-1}{x^4+9x^2+20}\,dx$Preview
  10. Q158Evaluate: $\int \frac{x^2+3}{(x^2-1)(x^2-2)}\,dx$Preview
  11. Q159Evaluate: $\int \frac{2x}{(2+x^2)(3+x^2)}\,dx$Preview
  12. Q160Evaluate: $\int \frac{2^x}{(4^x-3)(2^x-4)}\,dx$Preview
  13. Q161Evaluate: $\int \frac{3x-2}{(x+1)^2(x+3)}\,dx$Preview
  14. Q162Evaluate: $\int \frac{5x^2+20x+6}{x^3+2x^2+x}\,dx$Preview
  15. Q163Evaluate: $\int \frac{1}{x(1+4x^3+3x^6)}\,dx$Preview
  16. Q164Evaluate: $\int \frac{1}{x^3-1}\,dx$Preview
  17. Q165Evaluate: $\int \frac{(3\sin x-2)\cos x}{5-4\sin x-\cos^2 x}\,dx$Preview
  18. Q166Evaluate: $\int \frac{1}{\sin x+\sin 2x}\,dx$Preview
  19. Q167Evaluate: $\int \frac{1}{2\sin x+\sin 2x}\,dx$Preview
  20. Q168Evaluate: $\int \frac{1}{\sin 2x+\cos x}\,dx$Preview
  21. Q169Evaluate: $\int \frac{1}{\sin x(3+2\cos x)}\,dx$Preview
  22. Q170Evaluate: $\int \frac{5e^x}{(e^x+1)(e^{2x}+9)}\,dx$Preview
  23. Q171Evaluate: $\int \frac{2\log x+3}{x(3\log x+2)\left[(\log x)^2+1\right]}\,dx$Preview