Evaluate the following integrals: (2+3 ) x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\sin(2+3\log\text{x})}{\text{x}}\text{ dx}$

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Answer

$\int\frac{\sin(2+3\log\text{x})}{\text{x}}\text{ dx}$

Let $2+3\log\text{x}=\text{t}$

$\Rightarrow\frac{3}{\text{x}}=\frac{\text{dt}}{\text{dx}}$

$\Rightarrow\frac{\text{dx}}{\text{x}}=\frac{\text{dt}}{3}$

Now, $\int\frac{\sin(2+3\log\text{x})}{\text{x}}\text{ dx}$

$=\frac{1}{3}\int\sin\text{t dt}$

$=\frac{1}{3}[-\cos\text{t}]+\text{C}$

$=-\frac{1}{3}\cos(2+3\log\text{x})+\text{C}$

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