What is the integral of (3x+2)cos(5x) ?

The integral of (3x+2)cos(5x) is 3/25 (5xsin(5x)+cos(5x))+2/5 sin(5x)+C

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integral of (3x+2)cos(5x)

\int (3 x + 2) cos (5 x) d x

\frac{3}{25} (5 x sin (5 x) + cos (5 x)) + \frac{2}{5} sin (5 x) + C

Solution steps

\int (3 x + 2) cos (5 x) d x

Expand

(3 x + 2) cos (5 x) : 3 x cos (5 x) + 2 cos (5 x)

= \int 3 x cos (5 x) + 2 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 3 x cos (5 x) d x + \int 2 cos (5 x) d x

\int 3 x cos (5 x) d x = \frac{3}{25} (5 x sin (5 x) + cos (5 x))

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

= \frac{3}{25} (5 x sin (5 x) + cos (5 x)) + \frac{2}{5} sin (5 x)

Add a constant to the solution

= \frac{3}{25} (5 x sin (5 x) + cos (5 x)) + \frac{2}{5} sin (5 x) + C

y = x (x + 2) (x - 2), [- 2, 1]

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What is the integral of (3x+2)cos(5x) ?
The integral of (3x+2)cos(5x) is 3/25 (5xsin(5x)+cos(5x))+2/5 sin(5x)+C