Lesson 5 of 13 · 7 min
Using trigonometric identities
NCERT §7.3.2
∫sin²x dx has no helpful factor outside, so substitution stalls. An identity from Class 11 turns sin²x into something that integrates in one line.
The lesson in notes
In short
Squares, cubes and products of sines and cosines are rewritten with identities until each term is a single sine or cosine of a multiple angle.
Lowering the power: sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2.
Worked example: ∫cos²x dx = ½∫(1 + cos 2x) dx = x/2 + (sin 2x)/4 + C.
Products become sums: 2 sin A cos B = sin(A + B) + sin(A − B), 2 cos A cos B = cos(A + B) + cos(A − B), 2 sin A sin B = cos(A − B) − cos(A + B).
Worked example: sin 2x cos 3x = ½(sin 5x − sin x), so ∫sin 2x cos 3x dx = −(1/10) cos 5x + ½ cos x + C.
Cubes use the triple angle formula: sin³x = (3 sin x − sin 3x)/4, so ∫sin³x dx = −(3/4) cos x + (1/12) cos 3x + C. Writing sin³x = (1 − cos²x) sin x and substituting gives an answer that differs only by a constant.
Different correct methods can give answers that look different. They must differ by a constant; check by differentiating.