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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1 Answers
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
\(I=F(\frac{π}{2})-F(0)=\frac{1}{2} (\frac{π}{2}-\frac{sin⁡π}{2})-\frac{1}{2} (0-\frac{sin⁡0}{2})\)
=\(\frac{1}{4} (π-0)-0=\frac{π}{4}\).