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Q20   Find the integrals of the functions \frac{\cos 2x }{( \cos x + \sin x )^2}

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best_answer

we know that cos2x= cos^2x-sin^2x
therefore,

\frac{\cos 2x }{( \cos x + \sin x )^2}
\frac{\cos 2x}{1+\sin 2x}\\ \Rightarrow \int \frac{\cos 2x}{1+\sin 2x}\\  let 1+sin2x =t \Rightarrow 2cos2x\ dx = dt
Now the given integral can be written as

\therefore \int \frac{\cos 2x}{(\cos x+\sin x)^2}=\frac{1}{2}\int \frac{1}{t}dt
                                                 \\\Rightarrow \frac{1}{2}\log\left | t \right |+C\\ \Rightarrow \frac{1}{2}\log\left | 1+\sin 2x \right |+C\\=log|sin^2x+cos^2x+2sinxcosx|+C\\=\frac{1}{2}log|(sinx+cosx)^2|+C=log|sinx+cosx|+C

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manish

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