What is the integral of sin(t-s) ?

The integral of sin(t-s) is cos(t-s)+C

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integral of sin(t-s)

\int sin (t - s) d s

cos (t - s) + C

Solution steps

\int sin (t - s) d s

Apply u-substitution

: \int - sin (u) d u

= \int - sin (u) d u

Take the constant out:

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

= - \int sin (u) d u

Use the common integral:

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

= - (- cos (u))

Substitute back

u = t - s

= - (- cos (t - s))

Simplify

= cos (t - s)

Add a constant to the solution

= cos (t - s) + C

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n = 1 \sum \infty n (\frac{4}{3})^{n}