What is the integral of 3xsin(3x+1) ?

The integral of 3xsin(3x+1) is -xcos(3x+1)+1/3 sin(3x+1)+C

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integral of 3xsin(3x+1)

\int 3 x sin (3 x + 1) d x

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

Solution steps

\int 3 x sin (3 x + 1) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x sin (3 x + 1) d x

Apply Integration By Parts

= 3 (- \frac{1}{3} x cos (3 x + 1) - \int - \frac{1}{3} cos (3 x + 1) d x)

\int - \frac{1}{3} cos (3 x + 1) d x = - \frac{1}{9} sin (3 x + 1)

= 3 (- \frac{1}{3} x cos (3 x + 1) - (- \frac{1}{9} sin (3 x + 1)))

Simplify

3 (- \frac{1}{3} x cos (3 x + 1) - (- \frac{1}{9} sin (3 x + 1))) : - x cos (3 x + 1) + \frac{1}{3} sin (3 x + 1)

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

Add a constant to the solution

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

What is the integral of 3xsin(3x+1) ?
The integral of 3xsin(3x+1) is -xcos(3x+1)+1/3 sin(3x+1)+C