JEE Maths — Indefinite Integrals Complete Chapter Guide

Question

How do you decide which integration technique to apply for a given integral? Walk through the decision-making process with examples covering substitution, by parts, partial fractions, and trigonometric methods.

(JEE Main + Advanced — strategy + application)

Solution — Step by Step

Before starting any integral, ask these questions in order:

Is it a standard formula? ( $\int x^n\,dx$ , $\int e^x\,dx$ , $\int \sin x\,dx$ , etc.) — just apply directly.
Can a simple substitution clean it up? Look for a function and its derivative sitting together.
Is it a rational function? (polynomial/polynomial) — use partial fractions.
Is there a product of two different types? (like $x \cdot e^x$ or $x \cdot \sin x$ ) — use integration by parts.
Is it purely trigonometric? — use trig identities to simplify first.
Does it involve $\sqrt{a^2 - x^2}$ , $\sqrt{a^2 + x^2}$ , or $\sqrt{x^2 - a^2}$ ? — trigonometric or hyperbolic substitution.

If you see $f(g(x)) \cdot g'(x)$ , substitute $t = g(x)$ , so $dt = g'(x)\,dx$ .

Example: $\int \frac{2x}{1 + x^2}\,dx$

Let $t = 1 + x^2$ , so $dt = 2x\,dx$

\int \frac{dt}{t} = \ln|t| + C = \ln|1 + x^2| + C

For $\int u\,dv = uv - \int v\,du$ , choose $u$ by the ILATE priority:

Inverse trig ( $\sin^{-1}x$ , $\tan^{-1}x$ )
Logarithmic ( $\ln x$ )
Algebraic ( $x^n$ )
Trigonometric ( $\sin x$ , $\cos x$ )
Exponential ( $e^x$ )

The function higher in ILATE becomes $u$ ; the other becomes $dv$ .

Example: $\int x\,e^x\,dx$

$u = x$ (Algebraic), $dv = e^x\,dx$

= x\,e^x - \int e^x\,dx = x\,e^x - e^x + C = e^x(x - 1) + C

When the degree of numerator < degree of denominator, decompose:

\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}

Solving: $A = \frac{1}{3}$ , $B = -\frac{1}{3}$

\int \frac{dx}{(x-1)(x+2)} = \frac{1}{3}\ln|x-1| - \frac{1}{3}\ln|x+2| + C

Watch for these high-frequency patterns:

$\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C$ — the ” $e^x$ magic formula”
$\int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$
$\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C$

graph TD
    A["Given Integral"] --> B{"Standard formula?"}
    B -->|Yes| B1["Apply directly"]
    B -->|No| C{"f and f' together?"}
    C -->|Yes| C1["Substitution"]
    C -->|No| D{"Rational function?"}
    D -->|Yes| D1["Partial Fractions"]
    D -->|No| E{"Product of two types?"}
    E -->|Yes| E1["Integration by Parts"]
    E -->|No| F{"Trig expression?"}
    F -->|Yes| F1["Trig Identities"]
    F -->|No| G{"Square root of quadratic?"}
    G -->|Yes| G1["Trig Substitution"]
    style A fill:#fbbf24,stroke:#000,stroke-width:2px
    style C1 fill:#86efac,stroke:#000
    style E1 fill:#93c5fd,stroke:#000

Why This Works

Integration is essentially “reverse differentiation” — but unlike differentiation (which is algorithmic), integration requires pattern recognition. The decision flowchart above converts this pattern recognition into a systematic process.

The key insight: most JEE integrals are designed to test whether you can recognise the right technique, not whether you can execute complicated algebra. Once you pick the correct method, the actual computation is usually straightforward.

Common Mistake

The most frequent JEE error: students try integration by parts when a simple substitution would work. By parts is overkill for $\int x\sqrt{1+x^2}\,dx$ — just substitute $t = 1 + x^2$ . Always check substitution FIRST before jumping to by parts or partial fractions. The simpler technique saves time and reduces errors.

For JEE Advanced, master the $e^x[f(x) + f'(x)]$ pattern. About 1 question per paper uses this trick. If you see $e^x$ multiplied by the sum of a function and its derivative, the answer is simply $e^x f(x) + C$ . Recognising this instantly saves 3-4 minutes.