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

1 Answer
Wataru
Sep 8, 2014

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

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