Evaluate : 1 4 x^2+11 d x - Vidyadip

Question

Evaluate : $\int \frac{1}{4 x^2+11} \cdot d x$

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Answer

$
\begin{aligned}
& \text { Solution : I }=\int \frac{1}{4\left(x^2+\frac{11}{4}\right)} \cdot d x \\
& =\frac{1}{4} \cdot \int \frac{1}{x^2+\left(\frac{\sqrt{11}}{2}\right)^2} \cdot d x \\
& \because \quad \int \frac{1}{x^2+a^2} \cdot d x=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+c \\
& \mathrm{I}=\frac{1}{4} \cdot\left[\frac{1}{\left(\frac{\sqrt{11}}{2}\right)}\right] \cdot \tan ^{-1}\left[\frac{x}{\left(\frac{\sqrt{11}}{2}\right)}\right]+c \\
& =\frac{1}{2 \sqrt{11}} \tan ^{-1}\left(\frac{2 x}{\sqrt{11}}\right)+c \\
\end{aligned}
$

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