Evaluate the following intregals: 1 2x+3 ^2x dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{\cos2\text{x}+3\sin^2\text{x}}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{1}{\cos2\text{x}+3\sin^2\text{x}}\ \text{dx}$
$=\int\frac{1}{2\cos^2\text{x}-1+3\sin^2\text{x}}\ \text{dx}$
$\text{I}=\frac{1}{\sqrt{2}}\int\frac{1}{1+\text{t}^2}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\text{I}=\int\frac{\sec^2\text{x}}{2-\sec^2\text{x}+3\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{2-(1+\tan^2\text{x})^2+3\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{2-1-\tan^2\text{x}+3\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\text{dt}}{1+2\tan^2\text{x}}$
Let $\sqrt{2}\tan\text{x}=\text{t}$
$\sqrt{2}\sec^2\text{x dx}=\text{dt}$
$=\frac{1}{\sqrt{2}}\tan^{-1}\text{t}+\text{C}$
$\text{I}=\frac{1}{\sqrt{2}}\tan^{-1}(\sqrt{2}\tan\text{x})+\text{C}$

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