Question

How do you use Integration by Substitution to find `intcos^3(x)*sin(x)dx`?

Answer 1

By substitution,
`int cos^3xsin x dx=-1/4cos^4x+C`

Let `u=cosx`.
By taking the derivative,
`{du}/{dx}=-sinx`
By taking the reciprocal,
`{dx}/{du}=1/{-sinx}`
By multiplying by `du`,
`dx={du}/{-sinx}`

By rewriting the integral in terms of `u`,
#intcos^3xsinx dx =int u^3 sinx cdot {du}/{-sinx}#
by cancelling out `sinx`'s,
`=-int u^3 du`
by Power Rule,
`=-u^4/4+C`
by putting $u=cosx$ back in,
`=-1/4cos^4x+C`