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CBSE · Class 12 · Mathematics

Integration Using Trigonometric Identities (Ch 7: 7.3.2)

2 concepts · 16 practice questions with answers · Free to read

Squares and double-angle identities

From daily life

A tall box is easier to pack when you cut it into flat layers. Squares of sine and cosine are cut into simple cosines of a double angle.

Simple definition

The integral of sin²x or cos²x is not a standard result. Use the identities cos 2x = 1 − 2 sin²x = 2 cos²x − 1 to turn the square into a first power, which integrates easily.

Formula: sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2, sin x cos x = (sin 2x)/2

Rule 1: Reduce the power — Replace a square by a double-angle cosine.

  • Easy: ∫ cos²x dx = ∫ (1 + cos 2x)/2 dx = x/2 + (sin 2x)/4 + C.
  • Medium: ∫ sin²x dx = x/2 − (sin 2x)/4 + C.
  • Exam-type: ∫ 1 / (1 + cos 2x) dx = ∫ 1 / (2 cos²x) dx = (tan x)/2 + C.

Rule 2: Odd powers — Keep one factor and use sin²x + cos²x = 1.

  • Easy: ∫ sin³x dx = ∫ (1 − cos²x) sin x dx = −cos x + cos³x / 3 + C.
  • Medium: ∫ cos³x dx = sin x − sin³x / 3 + C.
  • Exam-type: ∫ (1 − cos 2x)/(1 + cos 2x) dx = ∫ tan²x dx = tan x − x + C.
Remember

A square of sine or cosine becomes a double angle. An odd power keeps one factor aside.

Common mistake

Writing ∫ cos²x dx = sin²x / 2 or (sin x)³ / 3.

Memory clue

Square down to double angle, odd power peels one off.

Practice questions

Easy
  1. 1Find ∫ cos²x dx.
  2. 2Find ∫ sin²x dx.
  3. 3Find ∫ sin 2x dx using sin 2x = 2 sin x cos x for a check of your earlier answer.
Medium
  1. 4Find ∫ sin³x dx.
  2. 5Find ∫ cos³x dx.
  3. 6Find ∫ 1/(1 + cos 2x) dx.
Challenging
  1. 7Find ∫ (1 − cos 2x)/(1 + cos 2x) dx.
  2. 8Find ∫ cot²x dx.
Show answers (try them yourself first)
  1. 1x/2 + (sin 2x)/4 + C
  2. 2x/2 − (sin 2x)/4 + C
  3. 3∫ 2 sin x cos x dx = sin²x + C, which equals −cos 2x / 2 + a different constant
  4. 4−cos x + cos³x / 3 + C
  5. 5sin x − sin³x / 3 + C
  6. 6(tan x) / 2 + C
  7. 7tan x − x + C
  8. 8∫ (cosec²x − 1) dx = −cot x − x + C

Product of sine and cosine to a sum

From daily life

Two waves played together can be written as one sum of waves. Products of sines and cosines of different angles also turn into sums.

Simple definition

Use 2 sin A cos B = sin(A + B) + sin(A − B), 2 cos A cos B = cos(A + B) + cos(A − B) and 2 sin A sin B = cos(A − B) − cos(A + B). Each term on the right integrates directly.

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

Rule 1: Product to sum — Pick the matching identity and integrate term by term.

  • Easy: ∫ sin 3x cos 2x dx = ½ ∫ (sin 5x + sin x) dx = −cos 5x / 10 − cos x / 2 + C.
  • Medium: ∫ cos 3x cos x dx = ½ ∫ (cos 4x + cos 2x) dx = sin 4x / 8 + sin 2x / 4 + C.
  • Exam-type: ∫ sin x sin 3x dx = ½ ∫ (cos 2x − cos 4x) dx = sin 2x / 4 − sin 4x / 8 + C.
Remember

The coefficient 1/2 stays outside, and each angle is divided by its own coefficient.

Common mistake

Integrating each factor separately and multiplying the results.

Memory clue

Product becomes sum, then integrate.

Practice questions

Easy
  1. 1Write sin 3x cos 2x as a sum of two sines.
  2. 2Find ∫ sin 3x cos 2x dx.
  3. 3Find ∫ cos 3x cos x dx.
Medium
  1. 4Find ∫ sin x sin 3x dx.
  2. 5Find ∫ sin 5x cos 3x dx.
  3. 6Find ∫ cos 2x cos 4x dx.
Challenging
  1. 7Find ∫ sin 4x sin 2x dx.
  2. 8Find ∫ cos x sin 3x dx.
Show answers (try them yourself first)
  1. 1(sin 5x + sin x) / 2
  2. 2−cos 5x / 10 − cos x / 2 + C
  3. 3sin 4x / 8 + sin 2x / 4 + C
  4. 4sin 2x / 4 − sin 4x / 8 + C
  5. 5½ ∫ (sin 8x + sin 2x) dx = −cos 8x / 16 − cos 2x / 4 + C
  6. 6½ ∫ (cos 6x + cos 2x) dx = sin 6x / 12 + sin 2x / 4 + C
  7. 7½ ∫ (cos 2x − cos 6x) dx = sin 2x / 4 − sin 6x / 12 + C
  8. 8½ ∫ (sin 4x + sin 2x) dx = −cos 4x / 8 − cos 2x / 4 + C