Question

Find \(\int \frac{5 cos^2⁡x}{1+sin⁡x} dx\).

(a) -3(x+cos⁡x)+C

(b) 5(x+cos⁡x)+C

(c) 5(-x+sin⁡x)+C

(d) 5(x-cos⁡x)+C

This question was addressed to me in an interview for internship.

Enquiry is from Methods of Integration-2 in portion Integrals of Mathematics – Class 12

Answer 1

The correct choice is (b) 5(x+cos⁡x)+C

For explanation I would say: \(\int \frac{5 cos^2⁡x}{1+sin⁡x} dx=\int \frac{5(1-sin^2⁡x)}{1+sin⁡x}=5\int \frac{(1+sin⁡x)(1-sin⁡x)}{(1+sin⁡x)} dx\)

=5∫ (1-sin⁡x)dx

=5(x-(-cos⁡x))=5(x+cos⁡x)+C