What is the integral of 5zcos(8z) ?

The integral of 5zcos(8z) is 5/64 (8zsin(8z)+cos(8z))+C

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integral of 5zcos(8z)

\int 5 z cos (8 z) d z

\frac{5}{64} (8 z sin (8 z) + cos (8 z)) + C

Solution steps

\int 5 z cos (8 z) d z

Take the constant out:

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

= 5 \cdot \int z cos (8 z) d z

Apply u-substitution

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

= 5 \cdot \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

= 5 \cdot \frac{1}{64} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= 5 \cdot \frac{1}{64} (u sin (u) - \int sin (u) d u)

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

= 5 \cdot \frac{1}{64} (u sin (u) - (- cos (u)))

Substitute back

u = 8 z

= 5 \cdot \frac{1}{64} (8 z sin (8 z) - (- cos (8 z)))

Simplify

5 \cdot \frac{1}{64} (8 z sin (8 z) - (- cos (8 z))) : \frac{5}{64} (8 z sin (8 z) + cos (8 z))

= \frac{5}{64} (8 z sin (8 z) + cos (8 z))

Add a constant to the solution

= \frac{5}{64} (8 z sin (8 z) + cos (8 z)) + C

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What is the integral of 5zcos(8z) ?
The integral of 5zcos(8z) is 5/64 (8zsin(8z)+cos(8z))+C