What is the integral of xcos(6x) ?

The integral of xcos(6x) is 1/36 (6xsin(6x)+cos(6x))+C

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integral of xcos(6x)

\int x cos (6 x) d x

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

Solution steps

\int x cos (6 x) d x

Apply u-substitution

= \int \frac{u cos ( u )}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int u cos (u) d u

Apply Integration By Parts

= \frac{1}{36} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= \frac{1}{36} (u sin (u) - (- cos (u)))

Substitute back

u = 6 x

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

Simplify

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

Add a constant to the solution

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

What is the integral of xcos(6x) ?
The integral of xcos(6x) is 1/36 (6xsin(6x)+cos(6x))+C