What is the integral of sin(w(t-x)) ?

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

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integral of sin(w(t-x))

\int sin (w (t - x)) d x

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

Solution steps

\int sin (w (t - x)) d x

Apply u-substitution

: \int - sin (w u) d u

= \int - sin (w u) d u

Take the constant out:

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

= - \int sin (w u) d u

Apply u-substitution

: \int \frac{sin ( v )}{w} d v

= - \int \frac{sin ( v )}{w} d v

Take the constant out:

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

= - \frac{1}{w} \cdot \int sin (v) d v

Use the common integral:

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

= - \frac{1}{w} (- cos (v))

Substitute back

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

Simplify

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

Add a constant to the solution

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

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What is the integral of sin(w(t-x)) ?
The integral of sin(w(t-x)) is 1/w cos(w(t-x))+C