Integrate the function in exercise. x 3x - Vidyadip

Question

Integrate the function in exercise.
$\text{x} \ \sin3\text{x}$

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Answer

Let $\text{I}=\int\text{x}\sin3\text{x dx}$
Taking x as first function and x as second function and integrating by parts, we obtain.
$\text{I}=\text{x}\int\sin3\text{x dx}-\int\Bigg\{\Big(\frac{\text{d}}{\text{dx}}\text{x}\Big)\int\sin3\text{x dx}\Bigg\}$
$=\text{x}\Big(\frac{-\cos3\text{x}}{3}\Big)-\int1.\Big(\frac{-\cos3\text{x}}{3}\Big)\text{dx}$
$=\frac{-\text{x}\cos3\text{x}}{3}+\frac{1}{3}\int\cos3\text{x} \ \text{dx}$
$=\frac{-\text{x}\cos3\text{x}}{3}+\frac{1}{9}\sin3\text{x} \ \text{dx}+\text{C}$

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