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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»

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

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

=\(\frac{1}{4} (π-0)-0=\frac{π}{4}\).


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