Integrate \frac{2 cos⁡2x}{(cos⁡x-sin⁡x)^2}.

Category: QuestionsIntegrate \frac{2 cos⁡2x}{(cos⁡x-sin⁡x)^2}.
Editor">Editor Staff asked 11 months ago

Integrate \frac{2 cos⁡2x}{(cos⁡x-sin⁡x)^2}.
(a) -log⁡(1+2sin⁡2x)+C
(b) \frac{1}{4} log⁡(1-sin⁡2x)+C
(c) –\frac{1}{4} log⁡(1+cos⁡2x)+C
(d) -log⁡(1-sin⁡2x)+C
The question was asked in my homework.
This question is from Methods of Integration-2 in portion Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

1 Answers
Editor">Editor Staff answered 11 months ago

Right choice is (d) -log⁡(1-sin⁡2x)+C
The explanation is: \int \frac{2 cos⁡2x}{(cos⁡x-sin⁡x)^2} dx=\int \frac{2 cos⁡2x}{cos^2⁡x+sin^2⁡x-sin⁡2x} \,dx \,(∵2 cos⁡x sin⁡x=sin⁡2x)
=\int \frac{2 cos⁡2x}{1-sin⁡2x} dx
Let 1-sin⁡2x=t
Differentiating w.r.t x, we get
-2 cos⁡2x dx=dt
2 cos⁡2x dx=-dt
\int \frac{2 cos⁡2x}{(cos⁡x-sin⁡x)^2} dx=-\int \frac{dt}{t}
Replacing t with 1-sin⁡2x, we get
∴\int \frac{2 cos⁡2x}{(cos⁡x-sin⁡x)^2} dx=-log⁡(1-sin⁡2x)+C