What is the integral of (2x+3)sin(4x) ?

The integral of (2x+3)sin(4x) is 2(-1/4 xcos(4x)+1/16 sin(4x))-3/4 cos(4x)+C

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integral of (2x+3)sin(4x)

\int (2 x + 3) sin (4 x) d x

2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) - \frac{3}{4} cos (4 x) + C

Solution steps

\int (2 x + 3) sin (4 x) d x

Expand

(2 x + 3) sin (4 x) : 2 x sin (4 x) + 3 sin (4 x)

= \int 2 x sin (4 x) + 3 sin (4 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 sin (4 x) d x + \int 3 sin (4 x) d x

\int 2 x sin (4 x) d x = 2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

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

= 2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) - \frac{3}{4} cos (4 x)

Add a constant to the solution

= 2 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) - \frac{3}{4} cos (4 x) + C

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What is the integral of (2x+3)sin(4x) ?
The integral of (2x+3)sin(4x) is 2(-1/4 xcos(4x)+1/16 sin(4x))-3/4 cos(4x)+C