Definite Integrals Part 2 – Class 12 CBSE & NCERT Practice
Definite Integrals Part 2 contains 50 carefully selected practice problems for Class 12 Mathematics, designed around CBSE and NCERT examination patterns. The questions provide practice in important topics such as properties of definite integrals, trigonometric integrals, substitution, symmetry, modulus functions, inverse trigonometric functions, integration by parts, and standard definite integrals. This worksheet is useful for CBSE Class 12 Maths board exam preparation, NCERT revision, school examinations, and JEE Mathematics practice. Attempt each question yourself first and then reveal the answer to check your solution.
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Definite Integrals — Board Exam Practice
Fifty definite-integral practice questions selected from the provided NCERT/CBSE material. Attempt each problem first, then reveal the answer.
1
Definite Integral
Evaluate: \[\int_0^{\pi/2}\frac{dx}{1+\tan x}\]
Solution
Answer \[\frac{\pi}{4}\]
2
Definite Integral
Evaluate: \[\int_0^{\pi/2}\frac{dx}{1+\cot x}\]
Solution
Answer \[\frac{\pi}{4}\]
3
Trigonometric Integral
Evaluate: \[\int_0^{\pi/2}\frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}}\,dx\]
Solution
Answer \[\frac{\pi}{4}\]
4
Symmetry Property
Evaluate: \[\int_0^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx\]
Solution
Answer \[\frac{\pi}{4}\]
5
Symmetry Property
Evaluate: \[\int_0^{\pi/2}\frac{\sin^n x}{\sin^n x+\cos^n x}\,dx\]
Solution
Answer \[\frac{\pi}{4}\]
6
Substitution
Evaluate: \[\int_0^{\pi/2}\frac{dx}{1+\sqrt{\tan x}}\]
Solution
Answer \[\frac{\pi}{4}\]
7
Standard Form
Evaluate: \[\int_0^a\frac{dx}{x+\sqrt{a^2-x^2}}\]
Solution
Answer \[\frac{\pi}{4}\]
8
Improper Integral
Evaluate: \[\int_0^\infty\frac{\log x}{1+x^2}\,dx\]
Solution
Answer \[0\]
9
Logarithmic Integral
Evaluate: \[\int_0^1\frac{\log(1+x)}{1+x^2}\,dx\]
Solution
Answer \[\frac{\pi}{8}\log2\]
10
Rational Function
Evaluate: \[\int_0^\infty\frac{x}{(1+x)(1+x^2)^2}\,dx\]
Solution
Answer \[\frac{\pi}{8}\]
11
Trigonometric Integral
Evaluate: \[\int_0^\pi\frac{x\tan x}{\sec x\,\csc x}\,dx\]
Solution
Answer \[\frac{\pi^2}{4}\]
12
Integration by Parts
Evaluate: \[\int_0^\pi x\sin x\cos^4x\,dx\]
Solution
Answer \[\frac{\pi}{5}\]
13
Integration by Parts
Evaluate: \[\int_0^\pi x\sin^3x\,dx\]
Solution
Answer \[\frac{2\pi}{3}\]
14
Logarithmic Integral
Evaluate: \[\int_0^\pi x\log(\sin x)\,dx\]
Solution
Answer \[-\frac{\pi^2}{2}\log2\]
15
Integration by Parts
Evaluate: \[\int_0^\pi x\cos^2x\,dx\]
Solution
Answer \[\frac{\pi^2}{4}\]
16
Trigonometric Integral
Evaluate: \[\int_{\pi/6}^{\pi/3}\frac{dx}{1+\cot^{5/2}x}\]
Solution
Answer \[\frac{\pi}{12}\]
17
Symmetry Property
Evaluate: \[\int_0^{\pi/2}\frac{\tan^7x}{\tan^7x+\cot^7x}\,dx\]
Solution
Answer \[\frac{\pi}{4}\]
18
Radical Integral
Evaluate: \[\int_2^8\frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}}\,dx\]
Solution
Answer \[4\pi-10\]
19
Trigonometric Integral
Evaluate: \[\int_0^\pi x\sin x\cos^2x\,dx\]
Solution
Answer \[\frac{\pi}{4}\]
20
Odd Function
Evaluate: \[\int_{-\pi/2}^{\pi/2}x\cos^2x\,dx\]
Solution
Answer \[0\]
21
Even Function
Evaluate: \[\int_{-\pi/2}^{\pi/2}\sin^4x\,dx\]
Solution
Answer \[\frac{3\pi}{8}\]
22
Trigonometric Integral
Evaluate: \[\int_{-\pi}^{\pi}(3\sin x-2)^2\,dx\]
Solution
Answer \[13\pi\]
23
Logarithmic Integral
Evaluate: \[\int_1^2\log\left(\frac{2-x}{2+x}\right)\,dx\]
Solution
Answer \[2-\log\frac{27}{4}\]
24
Symmetry Property
Evaluate: \[\int_{-\pi}^{\pi}(1-x^2)\sin x\cos^2x\,dx\]
Solution
Answer \[0\]
25
Integration by Parts
Evaluate: \[\int_0^{\pi/2}x\sin x\cos x\,dx\]
Solution
Answer \[\frac{\pi}{8}\]
26
Trigonometric Integral
Evaluate: \[\int_0^{\pi/2}\frac{x}{1+\cos x}\,dx\]
Solution
Answer \[\frac{\pi}{2} - \log2\]
27
Substitution
Evaluate: \[\int_0^{\pi/2}\frac{\cos x}{1+\sin x}\,dx\]
Solution
Answer \[\log2\]
28
Substitution
Evaluate: \[\int_0^{\pi/2}\frac{\sin x}{1+\cos x}\,dx\]
Solution
Answer \[\log2\]
29
Integration by Parts
Evaluate: \[\int_0^{\pi/2}\frac{x\sin x}{1+\cos x}\,dx\]
Solution
Answer \[\frac{\pi}{2}\log2\]
30
Definite Integral
Evaluate: \[\int_0^{\pi/2}\frac{x}{3+2\sin x+\cos x}\,dx\]
Solution
Answer \[\frac{\pi}{4}\log2\]
31
Trigonometric Identity
Evaluate: \[\int_0^{\pi/2}\frac{\sin^2x}{1+\cos x}\,dx\]
Solution
Answer \[\frac{\pi}{2}-1\]
32
Trigonometric Identity
Evaluate: \[\int_0^{\pi/2}\frac{\cos^2x}{1+\sin x}\,dx\]
Solution
Answer \[\frac{\pi}{2}-1\]
33
Modulus Function
Evaluate: \[\int_0^1|x^2-2x|\,dx\]
Solution
Answer \[\frac23\]
34
Rational Function
Evaluate: \[\int_0^1\frac{1-x^2}{x^4+x^2+1}\,dx\]
Solution
Answer \[\frac{1}{2}\ln 3\]
35
Standard Integral
Evaluate: \[\int_0^1\frac{dx}{1+x+x^2}\]
Solution
Answer \[\frac{\pi}{3\sqrt3}\]
36
Logarithmic Integral
Evaluate: \[\int_0^1\frac{x}{1+x^2}\,dx\]
Solution
Answer \[\frac12\log2\]
37
Logarithmic Integral
Evaluate: \[\int_0^1\frac{1-x}{1+x}\,dx\]
Solution
Answer \[2\log2-1\]
38
Radical Integral
Evaluate: \[\int_0^1\frac{dx}{\sqrt{1+x}-\sqrt{x}}\]
Solution
Answer \[\frac43(\sqrt2)\]
39
Trigonometric Integral
Evaluate: \[\int_0^{\pi/2}\sin^3x\,dx\]
Solution
Answer \[\frac23\]
40
Trigonometric Integral
Evaluate: \[\int_0^{\pi/2}\cos^3x\,dx\]
Solution
Answer \[\frac23\]
41
Trigonometric Integral
Evaluate: \[\int_0^{\pi/2}\sin^4x\,dx\]
Solution
Answer \[\frac{3\pi}{16}\]
42
Trigonometric Integral
Evaluate: \[\int_0^{\pi/2}\cos^4x\,dx\]
Solution
Answer \[\frac{3\pi}{16}\]
43
Modulus Function
Evaluate: \[\int_0^\pi|\cos x|\cos x\,dx\]
Solution
Answer \[0\]
44
Modulus Function
Evaluate: \[\int_0^{2\pi}|\sin x|\,dx\]
Solution
Answer \[4\]
45
Inverse Trigonometric
Evaluate: \[\int_0^1x(\cos^{-1}x)^2\,dx\]
Solution
Answer \[\frac{\pi^2 -4 }{16}\]
46
Inverse Trigonometric
Evaluate: \[\int_0^1(\cos^{-1}x)^2\,dx\]
Solution
Answer \[\pi-2\]
47
Trigonometric Integral
Evaluate: \[\int_0^{\pi/4}\tan^4x\,dx\]
Solution
Answer \[\frac{\pi}{4}-\frac23\]
48
Integration by Parts
Evaluate: \[\int_0^{\pi/2}x\sin x\,dx\]
Solution
Answer \[1\]
49
Polynomial Integral
Evaluate: \[\int_0^1(2x^2+3)\,dx\]
Solution
Answer \[\frac{11}{3}\]
50
Basic Definite Integral
Evaluate: \[\int_0^2 (x^2+2)\,dx\]
Solution
Integrate the polynomial term-by-term and substitute the limits.
Answer \[\frac{20}{3}\]
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