What is the integral of-5xcos(9x) ?

The integral of-5xcos(9x) is -5/81 (9xsin(9x)+cos(9x))+C

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integral of-5xcos(9x)

\int - 5 x cos (9 x) d x

- \frac{5}{81} (9 x sin (9 x) + cos (9 x)) + C

Solution steps

\int - 5 x cos (9 x) d x

Take the constant out:

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

= - 5 \cdot \int x cos (9 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{81} d u

= - 5 \cdot \int \frac{u cos ( u )}{81} d u

Take the constant out:

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

= - 5 \cdot \frac{1}{81} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= - 5 \cdot \frac{1}{81} (u sin (u) - \int sin (u) d u)

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

= - 5 \cdot \frac{1}{81} (u sin (u) - (- cos (u)))

Substitute back

u = 9 x

= - 5 \cdot \frac{1}{81} (9 x sin (9 x) - (- cos (9 x)))

Simplify

- 5 \cdot \frac{1}{81} (9 x sin (9 x) - (- cos (9 x))) : - \frac{5}{81} (9 x sin (9 x) + cos (9 x))

= - \frac{5}{81} (9 x sin (9 x) + cos (9 x))

Add a constant to the solution

= - \frac{5}{81} (9 x sin (9 x) + cos (9 x)) + C

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What is the integral of-5xcos(9x) ?
The integral of-5xcos(9x) is -5/81 (9xsin(9x)+cos(9x))+C