Integrate 2 sin^2⁡x+cos^2x.

Category: QuestionsIntegrate 2 sin^2⁡x+cos^2x.
Editor">Editor Staff asked 11 months ago

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
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1 Answers
Editor">Editor Staff answered 11 months ago

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