What is the integral of (6x+5)sin(7x) ?

The integral of (6x+5)sin(7x) is 6(-1/7 xcos(7x)+1/49 sin(7x))-5/7 cos(7x)+C

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integral of (6x+5)sin(7x)

\int (6 x + 5) sin (7 x) d x

6 (- \frac{1}{7} x cos (7 x) + \frac{1}{49} sin (7 x)) - \frac{5}{7} cos (7 x) + C

Solution steps

\int (6 x + 5) sin (7 x) d x

Expand

(6 x + 5) sin (7 x) : 6 x sin (7 x) + 5 sin (7 x)

= \int 6 x sin (7 x) + 5 sin (7 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 6 x sin (7 x) d x + \int 5 sin (7 x) d x

\int 6 x sin (7 x) d x = 6 (- \frac{1}{7} x cos (7 x) + \frac{1}{49} sin (7 x))

\int 5 sin (7 x) d x = - \frac{5}{7} cos (7 x)

= 6 (- \frac{1}{7} x cos (7 x) + \frac{1}{49} sin (7 x)) - \frac{5}{7} cos (7 x)

Add a constant to the solution

= 6 (- \frac{1}{7} x cos (7 x) + \frac{1}{49} sin (7 x)) - \frac{5}{7} cos (7 x) + C

What is the integral of (6x+5)sin(7x) ?
The integral of (6x+5)sin(7x) is 6(-1/7 xcos(7x)+1/49 sin(7x))-5/7 cos(7x)+C