Indefinite Integration – Complete Revision (JEE Mains & Advanced)
Indefinite Integration is not about memorising random formulas — it is about identifying forms. JEE strictly rotates questions around a fixed set of standard integrals and methods. This note covers the entire official syllabus with zero gaps.
1. Definition
\[
\int f(x)\,dx = F(x)+C \quad \text{where } \frac{dF}{dx}=f(x)
\]
2. ALL Standard Integrals (Must Memorise)
- \(\int x^n dx = \frac{x^{n+1}}{n+1}+C,\; n\neq-1\)
- \(\int \frac{1}{x}dx = \ln|x|+C\)
- \(\int e^x dx = e^x+C\)
- \(\int a^x dx = \frac{a^x}{\ln a}+C\)
- \(\int \sin x dx = -\cos x + C\)
- \(\int \cos x dx = \sin x + C\)
- \(\int \sec^2 x dx = \tan x + C\)
- \(\int \csc^2 x dx = -\cot x + C\)
- \(\int \sec x\tan x dx = \sec x + C\)
- \(\int \csc x\cot x dx = -\csc x + C\)
3. SIX IMPORTANT FORMS (JEE CORE)
Form 1: \(\int \frac{1}{x^2+a^2}dx\)
\[ = \frac{1}{a}\tan^{-1}\frac{x}{a}+C \]Form 2: \(\int \frac{1}{\sqrt{a^2-x^2}}dx\)
\[ = \sin^{-1}\frac{x}{a}+C \]Form 3: \(\int \frac{1}{\sqrt{x^2+a^2}}dx\)
\[ = \ln\left|x+\sqrt{x^2+a^2}\right|+C \]Form 4: \(\int \frac{1}{\sqrt{x^2-a^2}}dx\)
\[ = \ln\left|x+\sqrt{x^2-a^2}\right|+C \]Form 5: \(\int \frac{1}{x^2-a^2}dx\)
\[ = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C \]Form 6: \(\int \frac{1}{ax+b}dx\)
\[ = \frac{1}{a}\ln|ax+b|+C \]4. Methods of Integration (WITH EXAMPLES)
(a) Substitution Method
\[
\int x\cos(x^2)dx
\]
Let \(u=x^2\Rightarrow du=2x\,dx\)
\[
=\frac12\sin(x^2)+C
\]
(b) Integration by Parts
\[ \int u\,dv = uv - \int v\,du \] Example: \[ \int x\ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4}+C \](c) Partial Fractions
Used when denominator factorises. \[ \int \frac{1}{(x-1)(x+2)}dx \](d) Trigonometric Integrals
Use identities: \[ \sin^2x=\frac{1-\cos2x}{2} \](e) Special Type
\[ \int e^{ax}\sin bx\,dx \](f) Definite-to-Indefinite Conversion
Used in Advanced questions.5. JEE Traps (Very Important)
- Forgetting constant \(C\)
- Wrong inverse function (tan⁻¹ vs sin⁻¹)
- Ignoring absolute value in log
- Wrong substitution limits
- Missing domain restrictions
Final Truth:
Every JEE integration question is a disguise of these
6 forms + standard integrals + 4 methods.
Master recognition, not brute calculation.
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