What is the integral of (2x+1)sin(5x) ?

The integral of (2x+1)sin(5x) is -1/5 cos(5x)(2x+1)+2/25 sin(5x)+C

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integral of (2x+1)sin(5x)

\int (2 x + 1) sin (5 x) d x

- \frac{1}{5} cos (5 x) (2 x + 1) + \frac{2}{25} sin (5 x) + C

Solution steps

\int (2 x + 1) sin (5 x) d x

Apply Integration By Parts

: - \frac{1}{5} cos (5 x) (2 x + 1) - \int - \frac{2}{5} cos (5 x) d x

= - \frac{1}{5} cos (5 x) (2 x + 1) - \int - \frac{2}{5} cos (5 x) d x

\int - \frac{2}{5} cos (5 x) d x = - \frac{2}{25} sin (5 x)

= - \frac{1}{5} cos (5 x) (2 x + 1) - (- \frac{2}{25} sin (5 x))

Simplify

= - \frac{1}{5} cos (5 x) (2 x + 1) + \frac{2}{25} sin (5 x)

Add a constant to the solution

= - \frac{1}{5} cos (5 x) (2 x + 1) + \frac{2}{25} sin (5 x) + C

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What is the integral of (2x+1)sin(5x) ?
The integral of (2x+1)sin(5x) is -1/5 cos(5x)(2x+1)+2/25 sin(5x)+C