Evaluate the following intregals: 2 2+ ^22x dx

Question

Evaluate the following intregals:
$\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$
$\text{I}=\int\frac{2}{2+2\sin\text{x}\cos\text{x}}\ \text{dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\text{I}=\int\frac{\frac{1}{\cos^2\text{x}}}{\frac{1}{\cos^2\text{x}}+\frac{\sin\text{x}\cos\text{x}}{\cos^2\text{x}}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{\sec^2\text{x}+\tan\text{x}}\ \text{dx}$
$\text{I}=\int\frac{\sec^2\text{x}}{1+\tan^2\text{x}+\tan\text{x}}\ \text{dx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{x dx}=\text{dt}$
$\text{I}=\int\frac{\text{dt}}{\text{t}^2+\text{t}+1}$
$=\int\frac{\text{dt}}{\text{t}^2+2\text{t}\Big(\frac{1}{2}\Big)+\Big(\frac{1}{2}\Big)^2-\Big(\frac{1}{2}\Big)^2+1}$
$\text{I}=\int\frac{\text{dt}}{\Big(\text{t}+\frac{1}{2}\Big)^2+\Big(\frac{\sqrt{3}}{2}\Big)^2}$
$=\frac{1}{\frac{\sqrt{3}}{2}}\tan^{-1}\Bigg(\frac{\text{t}+\frac{1}{2}}{\frac{\sqrt{3}}{2}}\Bigg)+\text{C}$
$=\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{2\text{t}+1}{\sqrt{3}}\Big)+\text{C}$
$\text{I}=\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{2\tan\text{x}+1}{\sqrt{3}}\Big)+\text{C}$

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