What is the integral of-6xsin(6x+1) ?

The integral of-6xsin(6x+1) is xcos(6x+1)-1/6 sin(6x+1)+C

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integral of-6xsin(6x+1)

\int - 6 x sin (6 x + 1) d x

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

Solution steps

\int - 6 x sin (6 x + 1) d x

Take the constant out:

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

= - 6 \cdot \int x sin (6 x + 1) d x

Apply Integration By Parts

: - \frac{1}{6} x cos (6 x + 1) - \int - \frac{1}{6} cos (6 x + 1) d x

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

\int - \frac{1}{6} cos (6 x + 1) d x = - \frac{1}{36} sin (6 x + 1)

= - 6 (- \frac{1}{6} x cos (6 x + 1) - (- \frac{1}{36} sin (6 x + 1)))

Simplify

- 6 (- \frac{1}{6} x cos (6 x + 1) - (- \frac{1}{36} sin (6 x + 1))) : x cos (6 x + 1) - \frac{1}{6} sin (6 x + 1)

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

Add a constant to the solution

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

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What is the integral of-6xsin(6x+1) ?
The integral of-6xsin(6x+1) is xcos(6x+1)-1/6 sin(6x+1)+C