Integrate the function: x 4- ^ 2 x - Vidyadip

Question

Integrate the function: $\frac{\cos x}{\sqrt{4-\sin ^{2} x}}$

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Answer

Let $I=\frac{\cos x}{\sqrt{4-\sin ^{2} x}}$
Put sin x = t ⇒ cos x dx = dt
$\Rightarrow \int \frac{\cos x}{\sqrt{4-\sin ^{2} x}} d x=\int \frac{1}{\sqrt{4-t^{2}}} d t$
$=\int \frac{1}{\sqrt{\left(2^{2}-t^{2}\right)}} d t$
= $\sin ^{-1}\left(\frac{t}{2}\right)+C$
$\Rightarrow I=\sin ^{-1}\left(\frac{\sin x}{2}\right)+C$

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