Mathematics

Solve $$\int \dfrac{{\cos 2x}}{{{{\left( {\cos x + \sin x} \right)}^2}}} \ dx $$


SOLUTION

We have,

$$I=\int{\dfrac{\cos 2x}{{{\left( \cos x+\sin x \right)}^{2}}}}dx$$

$$ I=\int{\dfrac{\cos 2x}{{{\cos }^{2}}x+{{\sin }^{2}}x+2\sin x\cos x}}dx $$

$$ I=\int{\dfrac{\cos 2x}{1+\sin 2x}}dx $$

 

Let $$t=1+\sin 2x$$

$$ \dfrac{dt}{dx}=0+2\cos 2x $$

$$ \dfrac{dt}{2}=\cos 2xdx $$

 

Therefore,

$$ I=\dfrac{1}{2}\int{\dfrac{dt}{t}} $$

$$ I=\dfrac{1}{2}\ln \left( t \right)+C $$

 

Put the value of $$t$$, we get

$$I=\dfrac{1}{2}\ln \left( 1+\sin 2x \right)+C$$

 

Hence, this is the answer.

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Subjective Medium Published on 17th 09, 2020
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