Evaluate the following integrals: 4 ^2x-1 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\sin\text{x}}{\sqrt{4\cos^2\text{x}-1}}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{\sin\text{x}}{\sqrt{4\cos^2\text{x}-1}}\text{ dx}$
Let $2\cos\text{x}=\text{t}$
$\Rightarrow-2\sin\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin\text{x}\text{ dx}=-\frac{\text{dt}}{2}$
$\text{I}=-\frac{1}{2}\int\frac{\text{dt}}{\sqrt{\text{t}^2-1}}$
$=-\frac{1}{2}\log\Big|\text{t}+\sqrt{\text{t}^2-1}\Big|+\text{C}$ $\Big[\text{Since }\int\frac{1}{\sqrt{\text{x}^2-\text{a}^2}}\text{ dx}=\log\Big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\Big|+\text{C}\Big]$
$=-\frac{1}{2}\log\Big|2\cos\text{x}+\sqrt{4\cos^2\text{x}-1}\Big|+\text{C}$

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