Integration is one of the core skills in calculus, and the fastest way to get good at it is steady, focused practice. This page gives you 56 indefinite integral problems with full step-by-step solutions, arranged from the basic power rule all the way up to trigonometric, exponential, and logarithmic integrals. Try each question on your own first, then reveal the answer to check your work — and when you're done, you can print the whole set or save it as a PDF to revise offline.
What this integration worksheet covers
The problems are ordered by difficulty and grouped by type, so you can build up gradually or jump straight to the topic you're revising:
- Power rule integrals — the foundation of indefinite integration.
- Polynomials — integrating term by term.
- Radicals and fractional powers — such as √x and x2/3.
- Negative powers and the 1/x case — leading to the natural logarithm.
- Exponential integrals — including ex, eax, and ax.
- Trigonometric integrals — sin, cos, sec², csc², and more.
Basic rules of indefinite integration
Before you start, keep these core rules handy — almost every problem below uses one of them:
- Power rule: ∫ xn dx = xn+1 / (n+1) + C, for n ≠ −1.
- Constant rule: ∫ k dx = kx + C.
- Reciprocal rule: ∫ (1/x) dx = ln|x| + C.
- Exponential rule: ∫ ex dx = ex + C.
- Constant multiple & sum rules: pull constants out front and integrate each term separately.
Remember: every indefinite integral needs the constant of integration, + C. Leaving it out is the single most common mistake students make.
Who is this practice set for?
These indefinite integral questions are ideal for Class 12 / CBSE, A-Level Maths, and first-year college Calculus I students, as well as anyone preparing for exams like JEE or the AP Calculus test. Teachers can use the printable PDF as a ready-made homework sheet or in-class worksheet.
How to use this worksheet
Scroll down to the interactive worksheet below. Each problem has a Show solution button so you can attempt it first and then check your method, not just the final answer. When you're ready to study offline, click Print / Save as PDF — it exports only the questions and answers, ready to print or download.
Tags: #IndefiniteIntegrals #Integration #Calculus #MathPractice #IntegrationWorksheet #CalculusPractice #Maths #Class12Maths #APCalculus #IntegralCalculus #MathWorksheet #PrintableMath #StudyMaths #ExamPrep #JEEMaths #ALevelMaths #IntegrationRules #FreePDF
Class 12 · CBSE / NCERT · 56 Problems
Indefinite Integrals — Board Exam Practice
Fifty-six typical Class 12 integration questions from NCERT and CBSE board papers — solved using the \(\int \frac{f'}{f}=\log f\) rule, substitution, inverse-trig substitution, and partial fractions. Attempt each, then reveal the worked solution.
1
f′/f → log
Evaluate: \[\int \dfrac{\cos x}{2 + 3\sin x} \; dx\]
Solution
Numerator is (up to a constant) the derivative of the denominator.
Since \(\frac{d}{dx}(2+3\sin x)=3\cos x\), the result is \(\frac13\log|2+3\sin x|\).
Answer \(\dfrac{1}{3}\log|2 + 3\sin x| + C\)
2
f′/f → log
Evaluate: \[\int \dfrac{1 - \sin x}{x + \cos x} \; dx\]
Solution
\(\frac{d}{dx}(x+\cos x)=1-\sin x\), exactly the numerator.
Answer \(\log|x + \cos x| + C\)
3
f′/f → log
Evaluate: \[\int \dfrac{e^{3x}}{e^{3x} + 1} \; dx\]
Solution
\(\frac{d}{dx}(e^{3x}+1)=3e^{3x}\), so multiply by \(\tfrac13\).
Answer \(\dfrac{1}{3}\log(e^{3x} + 1) + C\)
4
f′/f → log
Evaluate: \[\int \dfrac{1}{x \log x} \; dx\]
Solution
Let \(t=\log x\), so \(dt=\frac1x dx\).
\(\int \frac{dt}{t}=\log|t|\).
Answer \(\log|\log x| + C\)
5
f′/f → log
Evaluate: \[\int \dfrac{1}{x(3 + \log x)} \; dx\]
Solution
Let \(t=3+\log x\), \(dt=\frac1x dx\).
Answer \(\log|3 + \log x| + C\)
6
f′/f → log
Evaluate: \[\int \dfrac{e^{x} + 1}{e^{x} + x} \; dx\]
Solution
\(\frac{d}{dx}(e^{x}+x)=e^{x}+1\), the numerator.
Answer \(\log|e^{x} + x| + C\)
7
f′/f → log
Evaluate: \[\int \dfrac{\sec^2 x}{\tan x + 2} \; dx\]
Solution
\(\frac{d}{dx}(\tan x+2)=\sec^2 x\).
Answer \(\log|\tan x + 2| + C\)
8
f′/f → log
Evaluate: \[\int \dfrac{\csc^2 x}{1 + \cot x} \; dx\]
Solution
\(\frac{d}{dx}(1+\cot x)=-\csc^2 x\), hence the minus sign.
Answer \(-\log|1 + \cot x| + C\)
9
f′/f → log
Evaluate: \[\int \dfrac{\cot x}{\log \sin x} \; dx\]
Solution
Let \(t=\log\sin x\), \(dt=\cot x\,dx\).
Answer \(\log|\log \sin x| + C\)
10
f′/f → log
Evaluate: \[\int \dfrac{1 + \cot x}{x + \log \sin x} \; dx\]
Solution
\(\frac{d}{dx}(x+\log\sin x)=1+\cot x\).
Answer \(\log|x + \log \sin x| + C\)
11
f′/f → log
Evaluate: \[\int \dfrac{1 + \tan x}{x + \log \sec x} \; dx\]
Solution
\(\frac{d}{dx}(x+\log\sec x)=1+\tan x\).
Answer \(\log|x + \log \sec x| + C\)
12
f′/f → log
Evaluate: \[\int \dfrac{\sec x \tan x}{3\sec x + 5} \; dx\]
Solution
\(\frac{d}{dx}(3\sec x+5)=3\sec x\tan x\).
Answer \(\dfrac{1}{3}\log|3\sec x + 5| + C\)
13
f′/f → log
Evaluate: \[\int \dfrac{1}{x \log x \,\log(\log x)} \; dx\]
Solution
Let \(t=\log(\log x)\); then \(dt=\frac{1}{x\log x}dx\).
Answer \(\log|\log(\log x)| + C\)
14
f′/f → log
Evaluate: \[\int \dfrac{10x^{9} + 10^{x}\log_e 10}{x^{10} + 10^{x}} \; dx\]
Solution
The numerator is exactly \(\frac{d}{dx}(x^{10}+10^{x})\).
Answer \(\log|x^{10} + 10^{x}| + C\)
15
f′/f → log
Evaluate: \[\int \dfrac{\sec x}{\log(\sec x + \tan x)} \; dx\]
Solution
\(\frac{d}{dx}\log(\sec x+\tan x)=\sec x\).
Let \(t=\log(\sec x+\tan x)\).
Answer \(\log|\log(\sec x + \tan x)| + C\)
16
f′/f → log
Evaluate: \[\int \dfrac{\csc x}{\log\!\left(\tan \frac{x}{2}\right)} \; dx\]
Solution
\(\frac{d}{dx}\log\tan\frac{x}{2}=\csc x\).
Answer \(\log\left|\log\!\left(\tan\frac{x}{2}\right)\right| + C\)
17
Power substitution
Evaluate: \[\int \sin^5 x \cos x \; dx\]
Solution
Let \(t=\sin x\), \(dt=\cos x\,dx\); integrate \(t^5\).
Answer \(\dfrac{\sin^6 x}{6} + C\)
18
Power substitution
Evaluate: \[\int \tan^{3/2} x \,\sec^2 x \; dx\]
Solution
Let \(t=\tan x\), \(dt=\sec^2 x\,dx\); integrate \(t^{3/2}\).
Answer \(\dfrac{2}{5}\tan^{5/2} x + C\)
19
Power substitution
Evaluate: \[\int x^3 \cos(x^4) \; dx\]
Solution
Let \(t=x^4\), \(dt=4x^3dx\).
Answer \(\dfrac{1}{4}\sin(x^4) + C\)
20
Power substitution
Evaluate: \[\int x^3 \sin(x^4 + 1) \; dx\]
Solution
Let \(t=x^4+1\), \(dt=4x^3dx\).
Answer \(-\dfrac{1}{4}\cos(x^4 + 1) + C\)
21
Power substitution
Evaluate: \[\int x\,e^{x^2} \; dx\]
Solution
Let \(t=x^2\), \(dt=2x\,dx\).
Answer \(\dfrac{1}{2}e^{x^2} + C\)
22
Log substitution
Evaluate: \[\int \dfrac{\log x}{x} \; dx\]
Solution
Let \(t=\log x\), \(dt=\frac1x dx\); integrate \(t\).
Answer \(\dfrac{(\log x)^2}{2} + C\)
23
Log substitution
Evaluate: \[\int \dfrac{(\log x)^2}{x} \; dx\]
Solution
Let \(t=\log x\); integrate \(t^2\).
Answer \(\dfrac{(\log x)^3}{3} + C\)
24
Log substitution
Evaluate: \[\int \dfrac{\sin(\log x)}{x} \; dx\]
Solution
Let \(t=\log x\), \(dt=\frac1x dx\).
Answer \(-\cos(\log x) + C\)
25
Log substitution
Evaluate: \[\int \dfrac{\sin(2 + 3\log x)}{x} \; dx\]
Solution
Let \(t=2+3\log x\), \(dt=\frac3x dx\).
Answer \(-\dfrac{1}{3}\cos(2 + 3\log x) + C\)
26
Power substitution
Evaluate: \[\int \cot^3 x \,\csc^2 x \; dx\]
Solution
Let \(t=\cot x\), \(dt=-\csc^2 x\,dx\); integrate \(-t^3\).
Answer \(-\dfrac{\cot^4 x}{4} + C\)
27
Substitution
Evaluate: \[\int \dfrac{1 + \cos x}{(x + \sin x)^3} \; dx\]
Solution
Let \(t=x+\sin x\), \(dt=(1+\cos x)dx\); integrate \(t^{-3}\).
Answer \(-\dfrac{1}{2(x + \sin x)^2} + C\)
28
Substitution
Evaluate: \[\int \dfrac{\sin x}{(1 + \cos x)^2} \; dx\]
Solution
Let \(t=1+\cos x\), \(dt=-\sin x\,dx\).
Answer \(\dfrac{1}{1 + \cos x} + C\)
29
Substitution
Evaluate: \[\int \dfrac{\sin 2x}{(a + b\cos 2x)^2} \; dx\]
Solution
Let \(t=a+b\cos 2x\), \(dt=-2b\sin 2x\,dx\).
Answer \(\dfrac{1}{2b\,(a + b\cos 2x)} + C\)
30
Substitution
Evaluate: \[\int 2x \sec^3(x^2+3)\tan(x^2+3) \; dx\]
Solution
Let \(t=x^2+3\); the integral becomes \(\int \sec^3 t\,\tan t\,dt=\tfrac13\sec^3 t\).
Answer \(\dfrac{1}{3}\sec^3(x^2 + 3) + C\)
31
Substitution
Evaluate: \[\int x^2 e^{x^3}\cos(e^{x^3}) \; dx\]
Solution
Let \(t=e^{x^3}\), \(dt=3x^2 e^{x^3}dx\).
Answer \(\dfrac{1}{3}\sin(e^{x^3}) + C\)
32
Substitution
Evaluate: \[\int e^{\cos^2 x}\sin 2x \; dx\]
Solution
Let \(t=\cos^2 x\), \(dt=-\sin 2x\,dx\).
Answer \(-e^{\cos^2 x} + C\)
33
Substitution
Evaluate: \[\int \cot x \,\log(\sin x) \; dx\]
Solution
Let \(t=\log\sin x\), \(dt=\cot x\,dx\); integrate \(t\).
Answer \(\dfrac{1}{2}(\log \sin x)^2 + C\)
34
Substitution
Evaluate: \[\int \sec x \,\log(\sec x + \tan x) \; dx\]
Solution
Let \(t=\log(\sec x+\tan x)\), \(dt=\sec x\,dx\).
Answer \(\dfrac{1}{2}\left[\log(\sec x + \tan x)\right]^2 + C\)
35
Substitution
Evaluate: \[\int \csc x \,\log(\csc x - \cot x) \; dx\]
Solution
Let \(t=\log(\csc x-\cot x)\), \(dt=\csc x\,dx\).
Answer \(\dfrac{1}{2}\left[\log(\csc x - \cot x)\right]^2 + C\)
36
Inverse-trig sub.
Evaluate: \[\int \dfrac{e^{m\sin^{-1}x}}{\sqrt{1 - x^2}} \; dx\]
Solution
Let \(t=m\sin^{-1}x\), \(dt=\frac{m}{\sqrt{1-x^2}}dx\).
Answer \(\dfrac{1}{m}e^{m\sin^{-1}x} + C\)
37
Inverse-trig sub.
Evaluate: \[\int \dfrac{e^{m\tan^{-1}x}}{1 + x^2} \; dx\]
Solution
Let \(t=m\tan^{-1}x\), \(dt=\frac{m}{1+x^2}dx\).
Answer \(\dfrac{1}{m}e^{m\tan^{-1}x} + C\)
38
Inverse-trig sub.
Evaluate: \[\int \dfrac{(\sin^{-1}x)^3}{\sqrt{1 - x^2}} \; dx\]
Solution
Let \(t=\sin^{-1}x\), \(dt=\frac{1}{\sqrt{1-x^2}}dx\); integrate \(t^3\).
Answer \(\dfrac{(\sin^{-1}x)^4}{4} + C\)
39
Inverse-trig sub.
Evaluate: \[\int \dfrac{1}{\sqrt{1 - x^2}\,(\sin^{-1}x)^2} \; dx\]
Solution
Let \(t=\sin^{-1}x\); integrate \(t^{-2}\).
Answer \(-\dfrac{1}{\sin^{-1}x} + C\)
40
Inverse-trig sub.
Evaluate: \[\int \dfrac{1}{\sqrt{\tan^{-1}x}\,(1 + x^2)} \; dx\]
Solution
Let \(t=\tan^{-1}x\); integrate \(t^{-1/2}\).
Answer \(2\sqrt{\tan^{-1}x} + C\)
41
Inverse-trig sub.
Evaluate: \[\int \dfrac{x \sin^{-1}(x^2)}{\sqrt{1 - x^4}} \; dx\]
Solution
Let \(t=\sin^{-1}(x^2)\), \(dt=\frac{2x}{\sqrt{1-x^4}}dx\).
Answer \(\dfrac{1}{4}\left(\sin^{-1}x^2\right)^2 + C\)
42
\(\sqrt{x}\) substitution
Evaluate: \[\int \dfrac{\sin\sqrt{x}}{\sqrt{x}} \; dx\]
Solution
Let \(t=\sqrt{x}\), \(dt=\frac{1}{2\sqrt{x}}dx\).
Answer \(-2\cos\sqrt{x} + C\)
43
\(\sqrt{x}\) substitution
Evaluate: \[\int \dfrac{\cos\sqrt{x}}{\sqrt{x}} \; dx\]
Solution
Let \(t=\sqrt{x}\).
Answer \(2\sin\sqrt{x} + C\)
44
\(\sqrt{x}\) substitution
Evaluate: \[\int \dfrac{\sec^2\sqrt{x}}{\sqrt{x}} \; dx\]
Solution
Let \(t=\sqrt{x}\); integrate \(2\sec^2 t\).
Answer \(2\tan\sqrt{x} + C\)
45
\(\sqrt{x}\) substitution
Evaluate: \[\int \dfrac{e^{\sqrt{x}}\cos\!\left(e^{\sqrt{x}}\right)}{\sqrt{x}} \; dx\]
Solution
Let \(t=e^{\sqrt{x}}\), \(dt=\frac{e^{\sqrt{x}}}{2\sqrt{x}}dx\).
Answer \(2\sin\!\left(e^{\sqrt{x}}\right) + C\)
46
Expand first
Evaluate: \[\int \dfrac{(1 + \sqrt{x})^2}{\sqrt{x}} \; dx\]
Solution
Expand: \(\frac{1+2\sqrt{x}+x}{\sqrt{x}}=x^{-1/2}+2+x^{1/2}\).
Integrate term by term.
Answer \(2\sqrt{x} + 2x + \dfrac{2}{3}x^{3/2} + C\)
47
Rational → arctan
Evaluate: \[\int \dfrac{e^{x}}{1 + e^{2x}} \; dx\]
Solution
Let \(t=e^{x}\), \(dt=e^{x}dx\); \(\int\frac{dt}{1+t^2}=\tan^{-1}t\).
Answer \(\tan^{-1}(e^{x}) + C\)
48
Rational → arctan
Evaluate: \[\int \dfrac{1}{e^{x} + e^{-x}} \; dx\]
Solution
Multiply by \(e^x/e^x\): \(\int\frac{e^x}{e^{2x}+1}dx\), then \(t=e^x\).
Answer \(\tan^{-1}(e^{x}) + C\)
49
Rational → arctan
Evaluate: \[\int \dfrac{3x^5}{1 + x^{12}} \; dx\]
Solution
Let \(t=x^6\), \(dt=6x^5dx\); \(\int\frac{(1/2)dt}{1+t^2}\).
Answer \(\dfrac{1}{2}\tan^{-1}(x^6) + C\)
50
Rational → arctan
Evaluate: \[\int \dfrac{\cos x}{\sin^2 x + 4\sin x + 5} \; dx\]
Solution
Let \(t=\sin x\); denominator \(=(t+2)^2+1\).
Answer \(\tan^{-1}(\sin x + 2) + C\)
51
Partial fractions
Evaluate: \[\int \dfrac{e^{x}}{(1 + e^{x})(2 + e^{x})} \; dx\]
Solution
Let \(t=e^x\); \(\frac{1}{(1+t)(2+t)}=\frac{1}{1+t}-\frac{1}{2+t}\).
Answer \(\log\left|\dfrac{1 + e^{x}}{2 + e^{x}}\right| + C\)
52
Partial fractions
Evaluate: \[\int \dfrac{e^{x}}{e^{2x} + 5e^{x} + 6} \; dx\]
Solution
Let \(t=e^x\); denominator \(=(t+2)(t+3)\).
Answer \(\log\left|\dfrac{e^{x} + 2}{e^{x} + 3}\right| + C\)
53
Trig identity
Evaluate: \[\int \sqrt{\dfrac{1 + \cos 2x}{1 - \cos 2x}} \; dx\]
Solution
\(\frac{1+\cos 2x}{1-\cos 2x}=\frac{2\cos^2 x}{2\sin^2 x}=\cot^2 x\).
So the integrand is \(|\cot x|\).
Answer \(\log|\sin x| + C\)
54
Trig identity
Evaluate: \[\int \dfrac{\cos x - \sin x}{1 + \sin 2x} \; dx\]
Solution
\(1+\sin 2x=(\sin x+\cos x)^2\); let \(t=\sin x+\cos x\).
\(dt=(\cos x-\sin x)dx\).
Answer \(-\dfrac{1}{\sin x + \cos x} + C\)
55
Trig identity
Evaluate: \[\int \dfrac{1 - \cot x}{1 + \cot x} \; dx\]
Solution
Multiply by \(\frac{\sin x}{\sin x}\): \(\frac{\sin x-\cos x}{\sin x+\cos x}\).
This is \(-\frac{d}{dx}\log(\sin x+\cos x)\).
Answer \(-\log|\sin x + \cos x| + C\)
56
Rational → arctan
Evaluate: \[\int \dfrac{x^2}{x^6 + a^6} \; dx\]
Solution
Let \(t=x^3\), \(dt=3x^2dx\); denominator \(=t^2+(a^3)^2\).
Answer \(\dfrac{1}{3a^3}\tan^{-1}\!\left(\dfrac{x^3}{a^3}\right) + C\)
0 Comments