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

Question

Evaluate the following integrals:
$\int\log\text{x}\frac{\sin\big\{1+(\log\text{x})^2\big\}}{\text{x}}\text{ dx}$

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Answer

$\int\log\text{x}\frac{\sin\big\{1+(\log\text{x})^2\big\}}{\text{x}}\text{ dx}$
Let $1+(\log\text{x})^2=\text{t}$
$\Rightarrow2\log\text{x}\times\frac{1}{\text{x}}\text{ dx}=\text{dt}$
$\Rightarrow\frac{\log\text{x}}{\text{x}}\text{ dx}=\frac{\text{dt}}{2}$
Now, $\int\log\text{x}\frac{\sin\big\{1+(\log\text{x})^2\big\}}{\text{x}}\text{ dx}$
$=\frac{1}{2}\int\sin(\text{t})\text{dt}$
$=\frac{1}{2}[-\cos\text{t}]+\text{C}$
$=-\frac{1}{2}\cos\big\{1+(\log\text{x})^2\big\}+\text{C}$

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