Indefinite Integration

Chapter 7: CBSE 2025-26 Compilation

1. Power & Exponential (Group 1)

Function Integral ($$ \int f(x) dx $$)
$$ x^n $$ ($$ n \neq -1 $$) $$ \frac{x^{n+1}}{n+1} + C $$
$$ \frac{1}{x} $$ $$ \log_e |x| + C $$
$$ e^x $$ $$ e^x + C $$
$$ a^x $$ $$ \frac{a^x}{\log_e a} + C $$

2. Trigonometric Integrals (Groups 2 & 3)

Function Integral ($$ + C $$)
$$ \sin x $$ $$ -\cos x + C $$
$$ \cos x $$ $$ \sin x + C $$
$$ \sec^2 x $$ $$ \tan x + C $$
$$ \csc^2 x $$ $$ -\cot x + C $$
$$ \sec x \tan x $$ $$ \sec x + C $$
$$ \csc x \cot x $$ $$ -\csc x + C $$
$$ \tan x $$ $$ \log |\sec x| + C $$
$$ \cot x $$ $$ \log |\sin x| + C $$
$$ \sec x $$ $$ \log |\sec x + \tan x| + C $$
$$ \csc x $$ $$ \log |\csc x – \cot x| + C $$

3. Inverse Trigonometric (Group 4)

Function Integral ($$ + C $$)
$$ \frac{1}{\sqrt{1-x^2}} $$ $$ \sin^{-1} x + C $$
$$ \frac{-1}{\sqrt{1-x^2}} $$ $$ \cos^{-1} x + C $$
$$ \frac{1}{1+x^2} $$ $$ \tan^{-1} x + C $$
$$ \frac{-1}{1+x^2} $$ $$ \cot^{-1} x + C $$
$$ \frac{1}{x\sqrt{x^2-1}} $$ $$ \sec^{-1} x + C $$
$$ \frac{-1}{x\sqrt{x^2-1}} $$ $$ \csc^{-1} x + C $$

4. Special Integrals (Groups 5 & 6)

Function Integral ($$ + C $$)
$$ \frac{1}{x^2 – a^2} $$ $$ \frac{1}{2a}\log \left|\frac{x-a}{x+a}\right| + C $$
$$ \frac{1}{a^2 – x^2} $$ $$ \frac{1}{2a}\log \left|\frac{a+x}{a-x}\right| + C $$
$$ \frac{1}{x^2 + a^2} $$ $$ \frac{1}{a}\tan^{-1}\frac{x}{a} + C $$
$$ \frac{1}{\sqrt{x^2 – a^2}} $$ $$ \log |x + \sqrt{x^2 – a^2}| + C $$
$$ \frac{1}{\sqrt{x^2 + a^2}} $$ $$ \log |x + \sqrt{x^2 + a^2}| + C $$
$$ \frac{1}{\sqrt{a^2 – x^2}} $$ $$ \sin^{-1}\frac{x}{a} + C $$
$$ \sqrt{a^2 – x^2} $$ $$ \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C $$
$$ \sqrt{x^2 – a^2} $$ $$ \frac{x}{2}\sqrt{x^2-a^2} – \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + C $$
$$ \sqrt{x^2 + a^2} $$ $$ \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + C $$
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