What is the integral of (2x-1)cos(5x) ?

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

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integral of (2x-1)cos(5x)

\int (2 x - 1) cos (5 x) d x

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

Solution steps

\int (2 x - 1) cos (5 x) d x

Expand

(2 x - 1) cos (5 x) : 2 x cos (5 x) - cos (5 x)

= \int 2 x cos (5 x) - cos (5 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 x cos (5 x) d x - \int cos (5 x) d x

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

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

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

Add a constant to the solution

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

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