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

Question

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

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Answer

Let $\text{I}=\int\frac{1}{3+2\cos^2\text{x}}\text{ dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\text{I}=\int\frac{\frac{1}{\cos^2\text{x}}}{\frac{3}{\cos^2\text{x}}+\frac{2\cos^2\text{x}}{\cos^2\text{x}}}$
$=\int\frac{\sec^2\text{x}}{2\sec^2\text{x}+2}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{3(1+\tan^2\text{x})+2}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{3+3\tan^2\text{x}+2}\text{ dx}$
$=\int\frac{\sec^2\text{x}}{5+3\tan^2\text{x}}\ \text{dx}$
Let $\sqrt{3}\tan\text{x}=\text{t}$
$\sqrt{3}\sec^2\text{x}\text{ dx}=\text{dt}$
$\text{I}=\frac{1}{\sqrt{3}}\int\frac{\text{dt}}{(\sqrt{5})^2+\text{t}^2}$
$=\frac{1}{\sqrt{3}+\sqrt{5}}\tan^{-1}\Big(\frac{\text{t}}{\sqrt{5}}\Big)+\text{C}$
$\text{I}=\frac{1}{\sqrt{15}}\tan^{-1}\Big(\frac{\sqrt{3}\tan\text{x}}{\sqrt{5}}\Big)+\text{C}$

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