# Evaluate the definite integral $$\int_0^1 sin^2⁡x \,dx$$.

Category: QuestionsEvaluate the definite integral $$\int_0^1 sin^2⁡x \,dx$$.
Editor">Editor Staff asked 11 months ago

Evaluate the definite integral $$\int_0^1 sin^2⁡x \,dx$$.

(a) –$$\frac{π}{2}$$

(b) π

(c) $$\frac{π}{4}$$

(d) $$\frac{π}{6}$$

This question was addressed to me by my school principal while I was bunking the class.

Asked question is from Fundamental Theorem of Calculus-2 topic in chapter Integrals of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

The correct answer is (c) $$\frac{π}{4}$$

To explain I would say: Let $$I=\int_0^{π/2}sin^{2⁡}x \,dx$$

F(x)=$$\int sin^2⁡x \,dx$$

=$$\int \frac{(1-cos⁡2x)}{2} \,dx$$

=$$\frac{1}{2} (x-\frac{sin⁡2x}{2})$$

Applying the limits, we get

$$I=F(\frac{π}{2})-F(0)=\frac{1}{2} (\frac{π}{2}-\frac{sin⁡π}{2})-\frac{1}{2} (0-\frac{sin⁡0}{2})$$

=$$\frac{1}{4} (π-0)-0=\frac{π}{4}$$.