Integration by Substitution Method: Class 12 Maths Guide

Integration by Substitution: Class 12 Maths Guide with Examples

Not every integral fits neatly into a standard formula. When the integrand looks complicated — a function nested inside another function, or a product where one part is the derivative of the other — the substitution method is usually your best tool. This technique, often called "change of variable," converts a tricky integral into a simpler one by temporarily replacing part of the expression with a new variable. This guide walks you through the logic, the standard results every CBSE Class 12 student should memorize, and plenty of solved examples.

What Is Integration by Substitution?

Suppose you want to evaluate I = ∫f(x) dx. If you substitute x = g(t), so that dx = g′(t) dt, the integral transforms into:

I = ∫f(g(t)) · g′(t) dt

The idea is to choose a substitution that turns a complicated function into one you can integrate directly using standard formulas. Once you integrate with respect to t, you substitute back to express the final answer in terms of x.

How to Spot the Right Substitution

The golden rule: look for a function whose derivative also appears (or almost appears) somewhere in the integrand. A few common patterns:

• If you see g(x) inside another function and g′(x) multiplying it, substitute t = g(x).
• If the integrand involves a composite function like sin(x² + 1), notice that the derivative of x² + 1 is 2x — so if 2x also appears, substitute t = x² + 1.
• For expressions involving √x, try t = √x.
• For expressions involving eˣ, log x, or inverse trig functions, these often combine neatly with their own derivatives.

Solved Examples Using Direct Substitution

Example 1: Evaluate ∫2x sin(x² + 1) dx

The derivative of x² + 1 is 2x, which already appears in the integrand. Let t = x² + 1, so dt = 2x dx.

∫sin t dt = −cos t + C = −cos(x² + 1) + C

Example 2: Evaluate ∫(4x + 2)√(x² + x + 1) dx

Let t = x² + x + 1, so dt = (2x + 1) dx. Rewrite 4x + 2 = 2(2x + 1):

= 2∫√t dt = 2 · (2/3)t^(3/2) + C = (4/3)(x² + x + 1)^(3/2) + C

Example 3: Evaluate ∫sin(tan⁻¹x) / (1 + x²) dx

Let t = tan⁻¹x, so dt = dx/(1 + x²).

= ∫sin t dt = −cos t + C = −cos(tan⁻¹x) + C

Standard Trigonometric Integrals Derived by Substitution

Four important results are derived using substitution and should be memorized for quick recall in exams:

• ∫tan x dx = log|sec x| + C — derived by writing tan x as sin x/cos x and substituting t = cos x
• ∫cot x dx = log|sin x| + C — substitute t = sin x
• ∫sec x dx = log|sec x + tan x| + C — multiply and divide by (sec x + tan x), then substitute t = sec x + tan x
• ∫cosec x dx = log|cosec x − cot x| + C — similarly, substitute t = cosec x + cot x

These four results appear frequently in board exam problems, especially when combined with other techniques like integration by parts.

Substitution with Multiple Steps

Some problems require substituting more than once. Consider:

Example 4: Evaluate ∫[tan⁴√x · sec²√x] / √x dx

First substitute t = √x, so dx/√x = 2dt. This gives 2∫tan⁴t sec²t dt. Now substitute u = tan t, so sec²t dt = du:

= 2∫u⁴ du = (2/5)u⁵ + C = (2/5)tan⁵t + C = (2/5)tan⁵√x + C

This two-step substitution shows why recognizing structure — rather than mechanically applying a rule — is the real skill being tested.

A Common Board Exam Trap: Recognize Before You Substitute

Many students try to force a substitution even when the integrand doesn't naturally support one. Before substituting, always check whether a standard formula or a trigonometric identity would solve the problem more directly. Substitution is powerful, but only when the derivative relationship genuinely exists in the integrand — forcing it usually leads to a dead end and wasted exam time.

Practice Problem for Self-Assessment

Try evaluating ∫cos x / [(1 + sin x)(2 + sin x)] dx on your own. Hint: substitute t = sin x first, and you'll need partial fractions afterward — a combination technique that's common in the trickier CBSE board problems (we cover partial fractions in detail in a separate guide).

Suggested Internal Links

• Link to: Indefinite Integrals Class 12: Formulas & Solved Examples (foundational formulas used in every substitution problem)
• Link to: Integration Using Trigonometric Identities (alternative approach for many trig-based integrals)
• Link to: Integration by Partial Fractions (needed for the practice problem above)
• Link to: Integration by Parts – Complete Guide (for products of functions)

Ready to Practice Substitution Problems?

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