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

The integral of (-3x+5)cos(2x) is -3/4 (2xsin(2x)+cos(2x))+5/2 sin(2x)+C

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

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

- \frac{3}{4} (2 x sin (2 x) + cos (2 x)) + \frac{5}{2} sin (2 x) + C

Solution steps

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

Expand

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

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

\int 3 x cos (2 x) d x = \frac{3}{4} (2 x sin (2 x) + cos (2 x))

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

= - \frac{3}{4} (2 x sin (2 x) + cos (2 x)) + \frac{5}{2} sin (2 x)

Add a constant to the solution

= - \frac{3}{4} (2 x sin (2 x) + cos (2 x)) + \frac{5}{2} sin (2 x) + C

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What is the integral of (-3x+5)cos(2x) ?
The integral of (-3x+5)cos(2x) is -3/4 (2xsin(2x)+cos(2x))+5/2 sin(2x)+C