Definite Integrals – 80 Practice Questions with Solutions
Master Definite Integrals for Class 12 Mathematics with this carefully prepared collection of 80 practice questions with solutions. This worksheet is designed for CBSE Class 12 Maths, NCERT preparation, board examinations, and competitive exam practice. The questions cover important concepts of definite integration, including properties of definite integrals, substitution, trigonometric integrals, integration by parts, inverse trigonometric functions, logarithmic functions, rational functions, symmetry, and standard integral formulas.
Each problem is accompanied by an answer so that students can first attempt the question independently and then verify their result. Regular practice with these Class 12 definite integral questions can improve calculation accuracy, conceptual understanding, problem-solving speed, and confidence in CBSE board exams. The worksheet is also useful for students preparing for JEE Main, JEE Advanced, CUET, and other mathematics entrance examinations.
Whether you are looking for NCERT definite integrals practice questions, CBSE Maths worksheets, solved integration problems, Class 12 Maths PDF resources, or board exam preparation material, this collection provides a structured way to revise and test your understanding of definite integrals. Attempt the questions without looking at the answers first, identify your weak areas, and then use the solutions for revision.
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Definite Integrals — Board Exam Practice
Eighty questions from the exact previous question-and-answer list. Attempt each, then reveal the answer.
1
Definite Integral
Evaluate: \[\int_{2}^{4}\frac{x}{x^2+1}\,dx\]
Answer \[\frac12\log\frac{17}{5}\]
2
Integration by Parts
Evaluate: \[\int_{0}^{\pi/2}x^2\sin x\,dx\]
Answer \[\pi-2\]
3
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\sin x\sin2x\,dx\]
Answer \[\frac23\]
4
Standard Form
Evaluate: \[\int_{0}^{2}\frac{dx}{\sqrt{3+2x-x^2}}\]
Answer \[\frac{\pi}{3}\]
5
Substitution
Evaluate: \[\int_{\pi/3}^{\pi/4}(\tan x+\cot x)^2\,dx\]
Answer \[-\frac{2}{\sqrt3}\]
6
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi}\frac{dx}{3+2\sin x+\cos x}\]
Answer \[\frac{\pi}{4}\]
7
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{\cos^2x}{1+3\sin^2x}\,dx\]
Answer \[\frac{\pi}{6}\]
8
Integration by Parts
Evaluate: \[\int_{0}^{\pi/2}x^2\cos2x\,dx\]
Answer \[-\frac{\pi}{4}\]
9
Substitution
Evaluate: \[\int_{4}^{12}x(x-4)^{1/3}\,dx\]
Answer \[\frac{720}{7}\]
10
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{\sin x\cos x}{1+\sin^4x}\,dx\]
Answer \[\frac{\pi}{8}\]
11
Exponential Integral
Evaluate: \[\int_{0}^{\frac12\ln3}\frac{e^x}{1+e^{2x}}\,dx\]
Answer \[\frac{\pi}{3}\]
12
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}(a^2\cos^2x+b^2\sin^2x)\,dx\]
Answer \[\frac{\pi}{4}(a^2+b^2)\]
13
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{dx}{5+4\sin x}\]
Answer \[\frac{2}{3}\tan^{-1}\left(\frac13\right)\]
14
Radical Function
Evaluate: \[\int_{0}^{\pi/2}\sqrt{1+\cos x}\,dx\]
Answer \[2\]
15
Inverse Trigonometric
Evaluate: \[\int_{0}^{1}\tan^{-1}x\,dx\]
Answer \[\frac{\pi}{4}-\frac12\log2\]
16
Trigonometric Integral
Evaluate: \[\int_{-\pi/4}^{\pi/4}\frac{dx}{1+\sin x}\]
Answer \[2\]
17
Definite Integral
Evaluate: \[\int_{0}^{\pi}\frac{\sin x}{\sin x+\cos x}\,dx\]
Answer \[\frac{\pi}{2}\]
18
Exponential Function
Evaluate: \[\int_{0}^{1}xe^{x^2}\,dx\]
Answer \[\frac{e-1}{2}\]
19
Radical Function
Evaluate: \[\int_{0}^{1}\sqrt{\frac{1-x}{1+x}}\,dx\]
Answer \[\frac{\pi}{2}-1\]
20
Substitution
Evaluate: \[\int_{0}^{a}\frac{x}{\sqrt{a^2+x^2}}\,dx\]
Answer \[a(\sqrt2-1)\]
21
Inverse Trigonometric
Evaluate: \[\int_{0}^{1}\tan^{-1}\left(\frac{2x}{1-x^2}\right)dx\]
Answer \[\frac{\pi}{2}-\log2\]
22
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\cos^4x\,dx\]
Answer \[\frac{3\pi}{16}\]
23
Modulus / Floor Function
Evaluate: \[\int_{0}^{2}2x[x]\,dx\]
Answer \[1\]
24
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi}\frac{dx}{1+\sin x}\]
Answer \[2\]
25
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi}\cos x\cos2x\,dx\]
Answer \[\frac{5}{12}\]
26
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\cos^3x\,dx\]
Answer \[\frac23\]
27
Rational Function
Evaluate: \[\int_{0}^{1}\frac{2x+3}{5x^2+1}\,dx\]
Answer \[\frac15\log6+\frac{3}{\sqrt5}\tan^{-1}\sqrt5\]
28
Trigonometric Identity
Evaluate: \[\int_{0}^{\pi}\left(\sin^2\frac{x}{2}-\cos^2\frac{x}{2}\right)dx\]
Answer \[0\]
29
Definite Integral
Evaluate: \[\int_{0}^{2}\frac{dx}{\sqrt{3+2x-x^2}}\]
Answer \[\frac{\pi}{3}\]
30
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{\cos^2x}{1+3\sin^2x}\,dx\]
Answer \[\frac{\pi}{6}\]
31
Logarithmic Integral
Evaluate: \[\int_{1}^{2}\frac{dx}{x(1+\log x)^2}\]
Answer \[\log2\]
32
Trigonometric Integral
Evaluate: \[\int_{\pi/6}^{\pi/4}\csc x\,dx\]
Answer \[\log(\sqrt2-1)-\log(2-\sqrt3)\]
33
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/6}\cos^{-3}2\theta\,\sin2\theta\,d\theta\]
Answer \[\frac34\]
34
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{dx}{2\cos x+4\sin x}\]
Answer \[\frac{1}{2\sqrt5}\log\left(\frac{\sqrt5+1}{\sqrt5-1}\right)\]
35
Radical Function
Evaluate: \[\int_{0}^{1}\frac{dx}{\sqrt{1+x}-\sqrt{x}}\]
Answer \[\frac43(\sqrt2-1)\]
36
Definite Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{\sin x\cos x}{1+\sin^4x}\,dx\]
Answer \[\frac{\pi}{8}\]
37
Trigonometric Identity
Evaluate: \[\int_{0}^{\pi/2}\frac{x}{\sin^2x+\cos^2x}\,dx\]
Answer \[\frac{\pi^2}{8}\]
38
Odd Function
Evaluate: \[\int_{-\pi}^{\pi}x^{10}\sin^7x\,dx\]
Answer \[0\]
39
Standard Trigonometric Form
Evaluate: \[\int_{0}^{\pi}\frac{dx}{6-\cos x}\]
Answer \[\frac{\pi}{\sqrt{35}}\]
40
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{dx}{2\cos x+4\sin x}\]
Answer \[\frac{1}{2\sqrt5}\log\left(\frac{\sqrt5+1}{\sqrt5-1}\right)\]
41
Trigonometric Integral
Evaluate: \[\int_{\pi/6}^{\pi/2}\frac{\csc x\cot x}{1+\csc^2x}\,dx\]
Answer \[\tan^{-1}\left(\frac13\right)\]
42
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\frac{dx}{4\cos x+2\sin x}\]
Answer \[\frac1{\sqrt5}\log\left(\frac{\sqrt5+1}{\sqrt5-1}\right)\]
43
Basic Integral
Evaluate: \[\int_0^1x\,dx\]
Answer \[\frac12\]
44
Beta-Type Integral
Evaluate: \[\int_0^1x(1-x)^5\,dx\]
Answer \[\frac1{42}\]
45
Polynomial Integral
Evaluate: \[\int_1^4(x^2+x)\,dx\]
Answer \[\frac{57}{2}\]
46
Exponential Function
Evaluate: \[\int_{-1}^{1}e^{2x}\,dx\]
Answer \[\frac{e^2-e^{-2}}{2}\]
47
Exponential Function
Evaluate: \[\int_2^3e^{-x}\,dx\]
Answer \[e^{-2}-e^{-3}\]
48
Polynomial Integral
Evaluate: \[\int_1^3(2x^2+5x)\,dx\]
Answer \[\frac{112}{3}\]
49
Polynomial Integral
Evaluate: \[\int_1^3(x^2+3x)\,dx\]
Answer \[\frac{62}{3}\]
50
Polynomial Integral
Evaluate: \[\int_0^2(x^2+2)\,dx\]
Answer \[\frac{20}{3}\]
51
Polynomial Integral
Evaluate: \[\int_0^3(x^2+1)\,dx\]
Answer \[12\]
52
Inverse Trigonometric
Evaluate: \[\int_0^1x(\tan^{-1}x)^2\,dx\]
Answer \[\frac{\pi^2}{16}-\frac{\pi}{4}+\frac12\log2\]
53
Inverse Trigonometric
Evaluate: \[\int_0^1(\cos^{-1}x)^2\,dx\]
Answer \[\pi-2\]
54
Trigonometric Integral
Evaluate: \[\int_0^{\pi/4}\tan^4x\,dx\]
Answer \[\frac{\pi}{4}-\frac23\]
55
Modulus Function
Evaluate: \[\int_1^3|x^2-2x|\,dx\]
Answer \[2\]
56
Definite Integral
Evaluate: \[\int_{0}^{\pi/2}x^2\sin x\,dx\]
Answer \[\pi-2\]
57
Definite Integral
Evaluate: \[\int_{0}^{\pi/2}\cos^2x\,dx\]
Answer \[\frac{\pi}{4}\]
58
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\cos^4x\,dx\]
Answer \[\frac{3\pi}{16}\]
59
Exponential / Trigonometric
Evaluate: \[\int_{0}^{\pi/6}\cos x\cos2x\,dx\]
Answer \[\frac5{12}\]
60
Polynomial Integral
Evaluate: \[\int_0^1\left(xe^{2x}+\sin\frac{\pi x}{2}\right)dx\]
Answer \[\frac{e^2}{4}+\frac14+\frac2\pi\]
61
Trigonometric Integral
Evaluate: \[\int_0^1x(1-x)^5\,dx\]
Answer \[\frac1{42}\]
62
Trigonometric Integral
Evaluate: \[\int_{0}^{\pi/2}\sin^3x\,dx\]
Answer \[\frac23\]
63
Logarithmic / Inverse Trigonometric
Evaluate: \[\int_{0}^{\pi/2}\frac{\cos x}{1+\sin^2x}\,dx\]
Answer \[\frac{\pi}{4}\]
64
Logarithmic Integral
Evaluate: \[\int_1^3\frac{\log x}{(x+1)^2}\,dx\]
Answer \[\frac34\log3-\log2\]
65
Logarithmic Integral
Evaluate: \[\int_0^{\pi/4}\frac{\tan^3x}{1+\cos2x}\,dx\]
Answer \[\frac18\]
66
Rational Function
Evaluate: \[\int_{0}^{\pi/2}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\,d\theta\]
Answer \[2(\sqrt2-1)\]
67
Rational Function
Evaluate: \[\int_0^1\frac{\tan^{-1}x}{1+x^2}\,dx\]
Answer \[\frac{\pi^2}{32}\]
68
Rational Function
Evaluate: \[\int_0^1\frac{1-x}{1+x}\,dx\]
Answer \[2\log2-1\]
69
Exponential Integral
Evaluate: \[\int_1^2\log x\,dx\]
Answer \[2\log2-1\]
70
Definite Integral
Evaluate: \[\int_1^2\frac{x+3}{x(x+2)}\,dx\]
Answer \[\frac12\log6\]
71
Exponential Integral
Evaluate: \[\int_1^2\frac{3x}{9x^2-1}\,dx\]
Answer \[\frac16(\log35-\log8)\]
72
Trigonometric Integral
Evaluate: \[\int_0^1\frac{dx}{2x^2+x+1}\]
Answer \[\frac{2}{\sqrt7}\left(\tan^{-1}\frac5{\sqrt7}-\tan^{-1}\frac1{\sqrt7}\right)\]
73
Definite Integral
Evaluate: \[\int_0^5x\sqrt{5-x}\,dx\]
Answer \[\frac{20\sqrt5}{3}\]
74
Definite Integral
Evaluate: \[\int_0^1\frac{2x}{1+x^4}\,dx\]
Answer \[\frac{\pi}{4}\]
75
Definite Integral
Evaluate: \[\int_1^3\frac{\cos(\log x)}{x}\,dx\]
Answer \[\sin(\log3)\]
76
Definite Integral
Evaluate: \[\int_0^{\pi/2}\frac{dx}{a^2\sin^2x+b^2\cos^2x}\]
Answer \[\frac{\pi}{2ab}\]
77
Definite Integral
Evaluate: \[\int_0^{\pi/2}\frac{dx}{a\cos x+b\sin x},\qquad a,b>0\]
Answer \[\frac{1}{\sqrt{a^2+b^2}}\log\left(\frac{a+b+\sqrt{a^2+b^2}}{a+b-\sqrt{a^2+b^2}}\right)\]
78
Definite Integral
Evaluate: \[\int_0^1\sqrt{x(1-x)}\,dx\]
Answer \[\frac{\pi}{8}\]
79
Definite Integral
Evaluate: \[\int_0^1\tan^{-1}x\,dx\]
Answer \[\frac{\pi}{4}-\frac12\log2\]
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