Evaluate : 1 3 x^2-7 d x - Vidyadip

Question

Evaluate : $\int \frac{1}{\sqrt{3 x^2-7}} \cdot d x$

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Answer

$
\begin{aligned}
& \text {I }=\int \frac{1}{\sqrt{3\left(x^2-\frac{7}{3}\right)} \cdot d x} \\
&=\int \frac{1}{\sqrt{3} \cdot \sqrt{x^2-\left(\frac{\sqrt{7}}{\sqrt{3}}\right)^2}} \cdot d x \\
&=\frac{1}{\sqrt{3}} \cdot \int \frac{1}{\sqrt{x^2-\left(\frac{\sqrt{7}}{\sqrt{3}}\right)^2}} \cdot d x \\
& \int \frac{1}{\sqrt{x^2-a^2}} \cdot d x=\log \left|x+\sqrt{x^2-a^2}\right|+c \\
& \mathrm{I}=\frac{1}{\sqrt{3}} \cdot \log \left(x+\sqrt{x^2-\left(\frac{\sqrt{7}}{\sqrt{3}}\right)^2}\right)+c \\
&=\frac{1}{\sqrt{3}} \cdot \log \left(x+\sqrt{x^2-\frac{7}{3}}\right)+c
\end{aligned}
$

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