Evalute the following integrals: 1- 2x x+ ^2x dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{1-\sin2\text{x}}{\text{x}+\cos^2\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\frac{1-\sin2\text{x}}{\text{x}+\cos^2\text{x}}\text{dx}$
Putting $\text{x}+\cos^2\text{x}=\text{t}$
$\Rightarrow1-2\cos\text{x}\times\sin\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow(1-\sin2\text{x})\text{dx}=\text{dt}$
$\therefore\text{I}=\int\frac{1}{\text{t}}\text{dt}$
$=\text{ln}|\text{t}|+\text{C}$
$=\text{ln}|\text{x}+\cos^2\text{x}|\text{C}\ \big[\because\text{t}=\text{x}+\cos^2\text{x}\big]$

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