Evaluate the integral I=∫_0^(π/2)sin^2x dx.

Evaluate the integral $I = \int_{0}^{π /2} sin^{2} x d x$ .

medium MCQ 3 marks Class 12 · Mathematics · Integrals

Answer

\frac{π}{4}

The correct answer is $\frac{π}{4}$ — Subtracting gives $I = \frac{π}{4} - 0 = \frac{π}{4}$ .

Use the half‑angle identity $sin^{2} x = \frac{1 - c o s 2 x}{2}$ .
Substitute into the integral: $I = \int_{0}^{π /2} \frac{1 - c o s 2 x}{2} d x = \frac{1}{2} \int_{0}^{π /2} 1 d x - \frac{1}{2} \int_{0}^{π /2} cos 2 x d x$ .
Integrate the first term: $\frac{1}{2} \int_{0}^{π /2} 1 d x = \frac{1}{2} [x]_{0}^{π /2} = \frac{π}{4}$ .
Integrate the second term: $\frac{1}{2} \int_{0}^{π /2} cos 2 x d x = \frac{1}{2} [\frac{s i n 2 x}{2}]_{0}^{π /2} = \frac{1}{4} (sin π - sin 0) = 0$ .
Subtracting gives $I = \frac{π}{4} - 0 = \frac{π}{4}$ .

Common mistake: Integrating

sin^{2} x

directly as

- cos x

instead of applying the half‑angle formula.

Concept tested: Definite integral using trigonometric identities