What is the integral of xcos(8x) ?

The integral of xcos(8x) is 1/64 (8xsin(8x)+cos(8x))+C

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integral of xcos(8x)

\int x cos (8 x) d x

\frac{1}{64} (8 x sin (8 x) + cos (8 x)) + C

Solution steps

\int x cos (8 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{64} d u

= \int \frac{u cos ( u )}{64} d u

Take the constant out:

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

= \frac{1}{64} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{64} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{64} (u sin (u) - (- cos (u)))

Substitute back

u = 8 x

= \frac{1}{64} (8 x sin (8 x) - (- cos (8 x)))

Simplify

= \frac{1}{64} (8 x sin (8 x) + cos (8 x))

Add a constant to the solution

= \frac{1}{64} (8 x sin (8 x) + cos (8 x)) + C

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6 x^{5}

What is the integral of xcos(8x) ?
The integral of xcos(8x) is 1/64 (8xsin(8x)+cos(8x))+C