What is the integral of sin(tx) ?

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

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integral of sin(tx)

\int sin (t x) d x

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

Solution steps

\int sin (t x) d x

Apply u-substitution

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

= \int \frac{sin ( u )}{t} d u

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = t x

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

Simplify

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

Add a constant to the solution

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

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