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

Question

Evaluate the following integrals:

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

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Answer

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

Let $\sin\text{x}=\text{t}$

$\Rightarrow\cos\text{x}\text{ dx}=\text{dt}$

Now, $\int\frac{\cos\text{x}\text{ dx}}{\sqrt{4+\sin^2\text{x}}}$

$=\int\frac{\text{dt}}{\sqrt{2^2-\text{t}^2}}$

$=\log\Big|\text{t}+\sqrt{4+\text{t}^2}\Big|+\text{C}$

$=\log\Big|\sin\text{x}+\sqrt{4+\sin^2\text{x}}\Big|+\text{C}$

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