Evalute the following integrals: 2x ( + )^2 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\cos2\text{x}}{(\cos\text{x}+\sin\text{x})^2}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\cos2\text{x}}{(\cos\text{x}+\sin\text{x})^2}\text{dx}$

$=\int\frac{\cos^2\text{x}-\sin^2\text{x}}{(\cos\text{x}+\sin\text{x})^2}\text{dx}$

$=\int\frac{\cos\text{x}-\sin\text{x}}{\cos\text{x}+\sin\text{x}}\text{dx}$

Putting $\cos\text{x}+\sin\text{x}=\text{t}$

$\Rightarrow-\sin\text{x}+\cos\text{x}=\frac{\text{dt}}{\text{dt}}$

$\Rightarrow(\cos\text{x}-\sin\text{x})\text{dx}=\text{dt}$

$\therefore\text{I}=\int\frac{1}{\text{t}}\text{dt}$

$=\text{ln}|\text{t}|+\text{C}$

$=\text{ln}|\cos\text{x}+\sin\text{x}|+\text{C}\ \big[\because\text{t}=\cos\text{x}+\sin\text{x}\big]$

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