Evaluate the following integrals:
Question:

Evaluate the following integrals:

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

Solution:

Let $\sin x=t$

Then $\mathrm{dt}=\cos \mathrm{x} \mathrm{dx}$

Hence, $\int \frac{\cos x}{\sqrt{4+\sin ^{2} x}} d x=\int \frac{d t}{\sqrt{2}^{2}+t^{2}}$

Since we have, $\left.\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left(x^{2}\right.}+a^{2}\right)\right]+c$

Therefore, $\int \frac{\mathrm{dt}}{\sqrt{2^{2}+\mathrm{t}^{2}}}=\log \left[\mathrm{t}+\sqrt{\mathrm{t}^{2}+2^{2}}\right]+\mathrm{c}$

$=\log \left[\mathrm{t}+\sqrt{\mathrm{t}^{2}+2^{2}}\right]+\mathrm{c}=\log \left[\sin \mathrm{x}+\sqrt{\sin ^{2} \mathrm{x}+4}\right]+\mathrm{c}$

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