What is the integral of-8xcos(7x) ?

The integral of-8xcos(7x) is -8/49 (7xsin(7x)+cos(7x))+C

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integral of-8xcos(7x)

\int - 8 x cos (7 x) d x

- \frac{8}{49} (7 x sin (7 x) + cos (7 x)) + C

Solution steps

\int - 8 x cos (7 x) d x

Take the constant out:

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

= - 8 \cdot \int x cos (7 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{49} d u

= - 8 \cdot \int \frac{u cos ( u )}{49} d u

Take the constant out:

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

= - 8 \cdot \frac{1}{49} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= - 8 \cdot \frac{1}{49} (u sin (u) - \int sin (u) d u)

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

= - 8 \cdot \frac{1}{49} (u sin (u) - (- cos (u)))

Substitute back

u = 7 x

= - 8 \cdot \frac{1}{49} (7 x sin (7 x) - (- cos (7 x)))

Simplify

- 8 \cdot \frac{1}{49} (7 x sin (7 x) - (- cos (7 x))) : - \frac{8}{49} (7 x sin (7 x) + cos (7 x))

= - \frac{8}{49} (7 x sin (7 x) + cos (7 x))

Add a constant to the solution

= - \frac{8}{49} (7 x sin (7 x) + cos (7 x)) + C

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What is the integral of-8xcos(7x) ?
The integral of-8xcos(7x) is -8/49 (7xsin(7x)+cos(7x))+C