What is the integral of cos(5y) ?

The integral of cos(5y) is 1/5 sin(5y)+C

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integral of cos(5y)

\int cos (5 y) d y

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

Solution steps

\int cos (5 y) d y

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{5} sin (u)

Substitute back

u = 5 y

= \frac{1}{5} sin (5 y)

Add a constant to the solution

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