What is the integral of sin(5t) ?

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

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integral of sin(5t)

\int sin (5 t) d t

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

Solution steps

\int sin (5 t) d t

Apply u-substitution

= \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 \int sin (u) d u

Use the common integral:

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

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

Substitute back

u = 5 t

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

Simplify

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

Add a constant to the solution

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