Evaluate the following intregals: 1 4 ^2x+3 ^2x dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{4\cos^2\text{x}+3\sin^2\text{x}}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{1}{4\cos^2\text{x}+3\sin^2\text{x}}\ \text{dx}$
Dividing numerator and demnominator by $\cos^2\text{x}$
$=\int\frac{\frac{1}{\cos^2\text{x}}}{4+9\tan^2\text{x}} \text {dx}$
$\text{I}=\int\frac{\sec^2\text{x}}{4+9\tan^2\text{x}}\ \text{dx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{x}\text{ dx}=\text{dt}$
$\text{I}=\int\frac{\text{dt}}{4+9(\text{t})^2}$
$=\int\frac{\text{dt}}{4+(3\text{t})^2}$
Let $3\text{t}=\text{u}$
$3\text{dt}=\text{du}$
$\text{I}=\frac{1}{3}\int\frac{\text{du}}{(2)^2+(\text{u})^2}$
$=\frac{1}{3}\times\frac{1}{2}\times\tan^{-1}\Big(\frac{\text{u}}{2}\Big)+\text{C}$
$\text{I}=\frac{1}{6}\tan^{-1}\Big(\frac{3\text{t}}{2}\Big)+\text{C}$
$\text{I}=\frac{1}{6}\tan^{-1}\Big(\frac{3\tan\text{x}}{2}\Big)+\text{C}$

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