What is the integral of (x-1)cos(5x) ?

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

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integral of (x-1)cos(5x)

\int (x - 1) cos (5 x) d x

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

Solution steps

\int (x - 1) cos (5 x) d x

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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