Evaluate the following integrals: 1 x 4-9( )^2 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{1}{\text{x}\sqrt{4-9(\log\text{x})^2}}\text{ dx}$

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Answer

$\int\frac{\text{dx}}{\text{x}\sqrt{4-9(\log\text{x})^2}}$
Let $\log\text{x}=\text{t}$
$\Rightarrow\frac{1}{\text{x}}\text{ dx}=\text{dt}$
Now, $\int\frac{\text{dx}}{\text{x}\sqrt{4-9(\log\text{x})^2}}$
$=\int\frac{\text{dt}}{\sqrt{4-9\text{t}^2}}$
$=\int\frac{\text{dt}}{\sqrt{2^2-(3\text{t})^2}}$
$=\frac{1}{3}\sin^{-1}\Big(\frac{3\text{t}}{2}\Big)+\text{C}$
$=\frac{1}{3}\sin^{-1}\Big(\frac{3\log\text{x}}{2}\Big)+\text{C}$

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