Which integration technique to use — substitution, parts, partial fractions, trig

Question

Given an integral, how do you decide which technique to use — substitution, integration by parts, partial fractions, or trigonometric substitution? Provide a decision framework with examples.

(JEE Main tests integration technique selection in 3-4 questions per session; CBSE 12 asks 2-3)

Solution — Step by Step

Look for a function and its derivative inside the integral. If you see $f(g(x)) \cdot g'(x)$ , substitute $u = g(x)$ .

Example: $\int 2x \cdot e^{x^2} dx$ → Let $u = x^2$ , $du = 2x\,dx$ → $\int e^u du = e^{x^2} + C$

When to use: When you spot a composite function and its inner function’s derivative nearby.

Use when the integral is a product of two different types of functions (polynomial times exponential, polynomial times trig, log times anything).

\int u\,dv = uv - \int v\,du

Choose $u$ using LIATE priority: Logarithmic > Inverse trig > Algebraic > Trigonometric > Exponential.

Example: $\int x\,e^x dx$ → $u = x$ , $dv = e^x dx$ → $xe^x - e^x + C$

Use when the integrand is a ratio of polynomials and the degree of numerator is less than the denominator.

First, factor the denominator completely. Then decompose:

Distinct linear factors: $\frac{A}{x-a} + \frac{B}{x-b}$
Repeated linear factors: $\frac{A}{x-a} + \frac{B}{(x-a)^2}$
Irreducible quadratic: $\frac{Ax+B}{x^2+bx+c}$

Example: $\int \frac{1}{x^2-1}dx = \int\left(\frac{1/2}{x-1} - \frac{1/2}{x+1}\right)dx$

Use when you see $\sqrt{a^2 - x^2}$ , $\sqrt{a^2 + x^2}$ , or $\sqrt{x^2 - a^2}$ .

$\sqrt{a^2 - x^2}$ : substitute $x = a\sin\theta$
$\sqrt{a^2 + x^2}$ : substitute $x = a\tan\theta$
$\sqrt{x^2 - a^2}$ : substitute $x = a\sec\theta$

flowchart TD
    A["Given integral"] --> B{See function and its derivative?}
    B -->|Yes| C["Substitution<br/>u = inner function"]
    B -->|No| D{Product of two types?}
    D -->|Yes| E["Integration by Parts<br/>LIATE rule"]
    D -->|No| F{Rational function?}
    F -->|Yes| G["Partial Fractions<br/>Factor denominator"]
    F -->|No| H{Square root of quadratic?}
    H -->|Yes| I["Trig Substitution"]
    H -->|No| J["Try algebraic manipulation<br/>or special formulas"]

Why This Works

Each technique handles a specific structure. Substitution simplifies composite functions. Parts handles products (by reducing one factor while integrating the other). Partial fractions break complex fractions into simple ones. Trig substitution eliminates square roots using Pythagorean identities.

The decision tree works because these structures are mutually recognisable — a rational function looks different from a product of exponential and polynomial, which looks different from a square root expression.

Alternative Method

Some integrals need multiple techniques in sequence. For example, $\int \frac{x}{(x^2+1)(x+1)}dx$ requires partial fractions first, then substitution for one of the resulting terms. Do not expect one technique to solve everything — sometimes you need to chain two or three together.

Common Mistake

Students jump to integration by parts for integrals that can be solved by simple substitution. For example, $\int x\cos(x^2)dx$ looks like a product requiring parts, but substituting $u = x^2$ solves it in one step. Always try substitution first — it is the fastest method and handles the majority of integrals. Reserve by parts for when substitution clearly does not work.