What is the integral of sin(8t) ?

The integral of sin(8t) is -1/8 cos(8t)+C

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integral of sin(8t)

\int sin (8 t) d t

- \frac{1}{8} cos (8 t) + C

Solution steps

\int sin (8 t) d t

Apply u-substitution

= \int sin (u) \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{8} (- cos (u))

Substitute back

u = 8 t

= \frac{1}{8} (- cos (8 t))

Simplify

= - \frac{1}{8} cos (8 t)

Add a constant to the solution

= - \frac{1}{8} cos (8 t) + C