Find the integral of the function 2 x ( x+ x)^ 2 - Vidyadip

Question

Find the integral of the function $\frac{\cos 2 x}{(\cos x+\sin x)^{2}}$

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Answer

Clearly, $\frac{\cos 2 x}{(\cos x+\sin x)^{2}}=\frac{\cos 2 x}{\cos ^{2} x+\sin ^{2} x+2 \sin x \cos x}=\frac{\cos 2 x}{1+\sin 2 x}$
Now, $\int \frac{\cos 2 x}{(\cos x+\sin x)^{2}} d x=\int \frac{\cos 2 x}{1+\sin 2 x} d x$
Let 1 + sin2x = t
$\Rightarrow$ 2cos2x dx = dt
Thus, $\int \frac{\cos 2 x}{(\cos x+\sin x)^{2}} d x=\frac{1}{2} \int \frac{1}{t} d t$
$= \frac{1}{2} \log |\mathrm{t}|+\mathrm{C}$
$= \frac{1}{2} \log |1+\sin 2 x|+C$
$= \frac{1}{2} \log \left|(\cos x+\sin x)^{2}\right|+C$
= log|sinx + cosx| + C

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