What is the integral of ycos(5y) ?

The integral of ycos(5y) is 1/25 (5ysin(5y)+cos(5y))+C

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integral of ycos(5y)

\int y cos (5 y) d y

\frac{1}{25} (5 y sin (5 y) + cos (5 y)) + C

Solution steps

\int y cos (5 y) d y

Apply u-substitution

: \int \frac{u cos ( u )}{25} d u

= \int \frac{u cos ( u )}{25} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= \frac{1}{25} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{25} (u sin (u) - (- cos (u)))

Substitute back

u = 5 y

= \frac{1}{25} (5 y sin (5 y) - (- cos (5 y)))

Simplify

= \frac{1}{25} (5 y sin (5 y) + cos (5 y))

Add a constant to the solution

= \frac{1}{25} (5 y sin (5 y) + cos (5 y)) + C

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What is the integral of ycos(5y) ?
The integral of ycos(5y) is 1/25 (5ysin(5y)+cos(5y))+C