What is the integral of (sin(5x))/5 ?

The integral of (sin(5x))/5 is -1/25 cos(5x)+C

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integral of (sin(5x))/5

\int \frac{sin ( 5 x )}{5} d x

- \frac{1}{25} cos (5 x) + C

Solution steps

\int \frac{sin ( 5 x )}{5} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int sin (5 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{5} d u

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

Use the common integral:

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

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

Substitute back

u = 5 x

= \frac{1}{5} \cdot \frac{1}{5} (- cos (5 x))

Simplify

\frac{1}{5} \cdot \frac{1}{5} (- cos (5 x)) : - \frac{1}{25} cos (5 x)

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

Add a constant to the solution

= - \frac{1}{25} cos (5 x) + C

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What is the integral of (sin(5x))/5 ?
The integral of (sin(5x))/5 is -1/25 cos(5x)+C