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

Question

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

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Answer

$\frac{\cos\text{x}}{\sqrt{4-\sin^{2}\text{x}}}$$\text{Let}\ \sin\text{x}=\text{t}\Rightarrow\cos\text{x}\ \text{dx}=\text{dt}$
$\Rightarrow\int\frac{\cos\text{x}}{\sqrt{4-\sin^{2}\text{x}}}\text{dx}=\int\frac{\text{dt}}{\sqrt{(2)^{2}-\text{(t)}^{2}}}$
$=\sin^{-1}\bigg(\frac{\text{t}}{2}\bigg)+\text{C}$
$=\sin^{-1}\bigg(\frac{\sin\text{X}}{2}\bigg)+\text{C}$

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