What is the integral of xsin(1+5x) ?

The integral of xsin(1+5x) is -1/5 xcos(1+5x)+1/25 sin(1+5x)+C

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integral of xsin(1+5x)

\int x sin (1 + 5 x) d x

- \frac{1}{5} x cos (1 + 5 x) + \frac{1}{25} sin (1 + 5 x) + C

Solution steps

\int x sin (1 + 5 x) d x

Apply Integration By Parts

: - \frac{1}{5} x cos (1 + 5 x) - \int - \frac{1}{5} cos (1 + 5 x) d x

= - \frac{1}{5} x cos (1 + 5 x) - \int - \frac{1}{5} cos (1 + 5 x) d x

\int - \frac{1}{5} cos (1 + 5 x) d x = - \frac{1}{25} sin (1 + 5 x)

= - \frac{1}{5} x cos (1 + 5 x) - (- \frac{1}{25} sin (1 + 5 x))

Simplify

= - \frac{1}{5} x cos (1 + 5 x) + \frac{1}{25} sin (1 + 5 x)

Add a constant to the solution

= - \frac{1}{5} x cos (1 + 5 x) + \frac{1}{25} sin (1 + 5 x) + C

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What is the integral of xsin(1+5x) ?
The integral of xsin(1+5x) is -1/5 xcos(1+5x)+1/25 sin(1+5x)+C