Question

Integrate 2 sin^2⁡x+cos^2x.

(a) \(\frac{3x}{2}+\frac{sin⁡2x}{4}+C\)

(b) \(\frac{3x}{2}-\frac{sin⁡2x}{4}+C\)

(c) \(\frac{x}{2}+\frac{sin⁡2x}{4}+C\)

(d) \(\frac{3x}{4}-\frac{2sin⁡2x}{2}+C\)

This question was addressed to me in a job interview.

My question comes from Methods of Integration-2 topic in portion Integrals of Mathematics – Class 12

Answer 1

The correct choice is (b) \(\frac{3x}{2}-\frac{sin⁡2x}{4}+C\)

Best explanation: \(\int \,2 \,sin^2⁡x +cos^2⁡x=\int sin^2⁡x+sin^2⁡x+cos^2⁡x dx\)

=\(\int sin^2⁡x+1 dx\)

=\(\int sin^2⁡x dx+\int 1 dx\)

=\(\int \frac{1-cos⁡2x}{2} dx+\int 1 dx\)

=\(\frac{x}{2}-\frac{sin⁡2x}{4}+x\)

=\(\frac{3x}{2}-\frac{sin⁡2x}{4}+C\)