Integration Using Trigonometric Identities: Class 12 Maths Guide

Integration Using Trigonometric Identities: Class 12 Maths Guide

Trigonometric integrands like sin²x, cos³x, or sin 3x cos 2x can't be integrated directly using standard formulas — none of the basic rules apply to powers or products of trigonometric functions in their raw form. The solution is to first rewrite the expression using a trigonometric identity, converting it into a form that standard formulas can handle. This is one of the most exam-favorite techniques in CBSE Class 12 Maths Chapter 7, and mastering it will save you significant time in both board exams and competitive tests like CUET.

Why We Need Identities Before Integrating

There is no direct formula for ∫sin²x dx or ∫cos³x dx in their original form. However, using identities like the power-reduction formulas or product-to-sum rules, these expressions can be rewritten as sums of simpler terms — each of which matches a standard integral. This "rewrite, then integrate" approach is the core strategy for this entire topic.

Power-Reduction Identities for sin²x and cos²x

Two identities you'll use constantly:

• cos 2x = 2cos²x − 1 ⟹ cos²x = (1 + cos 2x)/2
• cos 2x = 1 − 2sin²x ⟹ sin²x = (1 − cos 2x)/2

Example 1: Evaluate ∫cos²x dx

Using the identity: ∫(1 + cos 2x)/2 dx = (1/2)∫dx + (1/2)∫cos 2x dx = x/2 + (1/4)sin 2x + C

Example 2: Evaluate ∫sin²x dx

= (1/2)∫(1 − cos 2x)dx = x/2 − (1/4)sin 2x + C

Notice the sign difference between the two results — a common point of confusion that examiners test directly.

Integrating Odd Powers: sin³x and cos³x

For odd powers, a different strategy works better: split off one factor and convert the rest using the Pythagorean identity sin²x + cos²x = 1.

Example 3: Evaluate ∫sin³x dx

Using sin 3x = 3 sin x − 4 sin³x, we can write sin³x = (3 sin x − sin 3x)/4. Then:

= (3/4)∫sin x dx − (1/4)∫sin 3x dx = −(3/4)cos x + (1/12)cos 3x + C

Alternatively, this can be solved using substitution: write sin³x = sin²x · sin x = (1 − cos²x) sin x, let t = cos x, and integrate −∫(1 − t²)dt. Both methods give equivalent answers (though they may look different due to different constants of integration), and it's worth practicing both to see how they connect.

Product-to-Sum Formulas for Mixed Products

When the integrand is a product of sines and cosines with different arguments (like sin 2x cos 3x), direct integration is impossible. Instead, use the product-to-sum identity:

sin A cos B = (1/2)[sin(A+B) + sin(A−B)]

Example 4: Evaluate ∫sin 4x cos 3x dx

Using the identity: sin 4x cos 3x = (1/2)[sin 7x + sin x]

= (1/2)∫sin 7x dx + (1/2)∫sin x dx = −(1/14)cos 7x − (1/2)cos x + C

Similar identities exist for cos A cos B and sin A sin B — always check which pair of angles is involved before choosing the identity.

Integrating tan²x and cot²x

These use the Pythagorean identities sec²x − tan²x = 1 and cosec²x − cot²x = 1:

• tan²x = sec²x − 1, so ∫tan²x dx = tan x − x + C
• cot²x = cosec²x − 1, so ∫cot²x dx = −cot x − x + C

These conversions are quick but frequently forgotten under exam pressure — practice them until they're automatic.

A Combined Example: Higher Powers with Multiple Steps

Example 5: Evaluate ∫sin³x cos²x dx

Write sin³x = sin x (1 − cos²x), so the integrand becomes sin x (1 − cos²x) cos²x. Let t = cos x, dt = −sin x dx:

= −∫(1 − t²)t² dt = −∫(t² − t⁴)dt = −t³/3 + t⁵/5 + C = −(1/3)cos³x + (1/5)cos⁵x + C

This combines the "split an odd power" strategy with substitution — a pattern that appears often in 5-mark board questions.

Choosing the Right Identity: A Quick Decision Framework

1. Even powers of sin/cos alone (like sin²x, cos⁴x) → use power-reduction formulas
2. Odd powers of sin/cos alone → split one factor, use Pythagorean identity, then substitute
3. Products of sin/cos with different arguments → use product-to-sum formulas
4. tan²x or cot²x → convert using sec²x or cosec²x identities

Keeping this framework in mind during your revision will help you instantly recognize which identity a given problem requires — a skill that directly translates into faster, more accurate answers in timed exams.

Suggested Internal Links

• Link to: Indefinite Integrals Class 12: Formulas & Solved Examples (for the base formulas used here)
• Link to: Integration by Substitution Method (used alongside identities in combined problems)
• Link to: Integration by Parts – ILATE Rule & Examples (for products involving trig and algebraic functions)
• Link to: CBSE Class 12 Maths Question Bank (for extra practice)

Ready to Practice Trigonometric Integration?

Recognizing the right identity under exam pressure takes repeated practice, not just formula memorization. ChampionsPrep offers AI-powered doubt resolution and topic-wise practice questions at ₹10 per use, so you can drill this exact skill until it becomes second nature. Practice Trigonometric Integrals on ChampionsPrep →

Test Your Knowledge