Evaluate the following integrals: _ 1 ^3 ( ) x dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{1}^\limits{3}\frac{\cos(\log\text{x})}{\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int_{1}^\limits{3}\frac{\cos(\log\text{x})}{\text{x}}\text{ dx}$
Let $\log\text{x}=\text{t}$ Then, $\frac{1}{\text{x}}\text{ dx}=\text{dt}$
When $\text{x}=1,\text{t}=0$ and $\text{x}=3,\text{t}=\log3$
$\therefore\ \text{I}=\int_{0}^\limits{\log3}\cos\text{t dt}$
$=\big[\sin\text{t}\big]^{\log3}_0$
$=\sin(\log3)$

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