What is the integral of-4xsin(5x) ?

The integral of-4xsin(5x) is -4(-1/5 xcos(5x)+1/25 sin(5x))+C

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integral of-4xsin(5x)

\int - 4 x sin (5 x) d x

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

Solution steps

\int - 4 x sin (5 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 4 \cdot \int x sin (5 x) d x

Apply Integration By Parts

= - 4 (- \frac{1}{5} x cos (5 x) - \int - \frac{1}{5} cos (5 x) d x)

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

= - 4 (- \frac{1}{5} x cos (5 x) - (- \frac{1}{25} sin (5 x)))

Simplify

= - 4 (- \frac{1}{5} x cos (5 x) + \frac{1}{25} sin (5 x))

Add a constant to the solution

= - 4 (- \frac{1}{5} x cos (5 x) + \frac{1}{25} sin (5 x)) + C

What is the integral of-4xsin(5x) ?
The integral of-4xsin(5x) is -4(-1/5 xcos(5x)+1/25 sin(5x))+C