What is the integral of-25sin(-5t) ?

The integral of-25sin(-5t) is -5cos(5t)+C

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integral of-25sin(-5t)

\int - 25 sin (- 5 t) d t

- 5 cos (5 t) + C

Solution steps

\int - 25 sin (- 5 t) d t

Take the constant out:

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

= - 25 \cdot \int sin (- 5 t) d t

Apply u-substitution

= - 25 \cdot \int - \frac{1}{5} sin (u) d u

Take the constant out:

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

= - 25 (- \frac{1}{5} \cdot \int sin (u) d u)

Use the common integral:

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

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

Substitute back

u = - 5 t

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

Simplify

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

= - 5 cos (5 t)

Add a constant to the solution

= - 5 cos (5 t) + C