Question

Find \(\int_0^{\frac{π}{4}} \,9 \,cos^2⁡x \,dx\).

(a) \(\frac{9}{2}\left (\frac{π}{6}-1\right)\)

(b) \(\frac{9}{4}\left (\frac{π}{2}+1\right)\)

(c) \(\frac{9}{4}\left (\frac{π}{2}-1\right)\)

(d) \(\left (\frac{π}{2}-1\right)\)

The question was asked during an online interview.

Enquiry is from Fundamental Theorem of Calculus-1 in division Integrals of Mathematics – Class 12

Answer 1

Right answer is (c) \(\frac{9}{4}\left (\frac{π}{2}-1\right)\)

For explanation: Let I=\(\int_0^{\frac{π}{4}} \,9 \,cos^2⁡x \,dx\).

F(x)=\(\int \,9 \,cos^2⁡x \,dx\)

=9\(\int(\frac{1+cos⁡2x}{2})dx\)

=\(\frac{9}{2} (x-\frac{sin⁡2x}{2})\)

Applying the limits, we get

I=\(F(\frac{π}{4})-F(0)=\frac{9}{2} \left (\frac{π}{4}-\frac{sin⁡2(\frac{π}{4})}{2}\right)-\frac{9}{2} (0-\frac{sin⁡0}{2})\)

=\(\frac{9}{2}\left (\frac{π}{4}-\frac{sin⁡π/2}{2}\right )=\frac{9}{4} (π/2-1)\)