What is the integral of asin(bt) ?

The integral of asin(bt) is -a 1/b cos(bt)+C

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integral of asin(bt)

\int a sin (b t) d t

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

Solution steps

\int a sin (b t) d t

Take the constant out:

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

= a \cdot \int sin (b t) d t

Apply u-substitution

: \int \frac{sin ( u )}{b} d u

= a \cdot \int \frac{sin ( u )}{b} d u

Take the constant out:

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

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

Use the common integral:

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

= a \frac{1}{b} (- cos (u))

Substitute back

u = b t

= a \frac{1}{b} (- cos (b t))

Simplify

= - a \frac{1}{b} cos (b t)

Add a constant to the solution

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

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