Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Integrals
0 votes

Choose the correct answers in $\int\frac{\large \cos 2x}{\large (\sin x+\cos x)^2}dx$ is equal to

$\begin{array}{1 1} A\;\frac{-1}{\sin x+\cos x}+C \\ (B)\;log|\sin x+\cos x\;|+C \\ C\;log|\sin x-\cos x\;|+C \\D\;\frac{1}{(\sin x+\cos x)^2} \end{array} $

Can you answer this question?

1 Answer

0 votes
  • (i)In a function f(x) is substituted by t, then $f'(x)dx=dt$ then $\int f(x)dx=\int t.dt$
  • (ii) $ \cos 2x=\cos ^2 x-\sin ^2 x$
  • (iii) $\int \frac{dx}{x}=log |x|+c$
Given $I=\int \frac{\cos 2x}{(\cos x+\sin x )^2}dx$
But we know $\cos 2x=\cos ^2x-\sin ^2x$
Therefore $I=\int \frac{\cos ^2x -\sin ^2x}{(\cos x+\sin x)^2}$
$\cos ^2x-\sin ^2 x=(\cos x+\sin x)(\cos x-\sin x)$
Therefore $I=\int \frac{(\cos x+\sin x)(\cos x -\sin x)}{(\cos x+\sin x)^2}$
$=\int \frac{\cos x-\sin x}{\cos x+\sin x} dx$
Let $\cos x +\sin x=t;$ on differentiating w.r.t x we get
$(-\sin x+\cos x)dx=dt$
Substituting for t and dt
$I=\int \frac{dt}{t}$
on integrating we get
$ I=log |t|+c$
substituting for t we get
$I=log |\cos x +\sin x|+c$
Hence the correct answer is B



answered Mar 13, 2013 by meena.p
Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App