Evaluate the following integrals: ( ) x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\sin(\log\text{x})}{\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{\sin(\log\text{x})}{\text{x}}\text{ dx}\ ....(1)$

Let $\log\text{x}=\text{t}$ then,

$\text{d}(\log\text{x})=\text{dt}$

$\Rightarrow\frac{1}{\text{x}}\text{ dx}=\text{dt}$

Putting $\log\text{x}=\text{t}$ and $\frac{1}{\text{x}}\text{ dx}=\text{dt}$ in equation (1),

We get,

$\text{I}=\int\sin\text{t dt}$

$=-\cos\text{t}+\text{C}$

$=-\cos\big(\log\text{x}\big)+\text{C}$

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