What is the integral of 1/5 sin(5x) ?

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

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integral of 1/5 sin(5x)

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

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

Solution steps

\int \frac{1}{5} sin (5 x) 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

\frac{d}{d t} (- 4 t^{3})

\int - \frac{1}{8} sin (2 x) d x

x \to 1 lim (lo g_{4} (x))

x \to 8 + lim (\frac{1}{x - 8} - \frac{1}{x - 8})

x \to 4 lim (\frac{5 + x}{x - 4})

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