What is the integral of ycos(5y) ?

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

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

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

= \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

\int 3 - 2 x^{2} d x

\int \frac{x ^{3}}{( 1 + x ^{8} ) ^{2}} d x

\int \frac{3}{x ( ln ( x ) ) ^{4}} d x

\int 2 (x^{2} + x + 2) e^{- 2 x} d x

in t e g r a l cos^{3} (x)

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