Question

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

Answer 1

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

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

=\(\frac{1}{4} (π-0)-0=\frac{π}{4}\).