Question

What parts do I need to integrate sin^(3x)*cos^2(x)dx? (Do I need to use algebraic substitution? If so what is the u value?)

Answer 1

\[\int\limits \sin ^3x \cos ^2x ~dx=\int\limits \sin x ~\sin ^2x \cos ^2xdx=\int\limits \cos ^2x \left( 1-\cos ^2x \right)\sin x dx\]
\[=\int\limits \cos ^2x \sin x~dx-\int\limits \cos ^4x \sin x~dx\]
put\[\cos x=t,\]
diff.
\[-\sin x~dx=dt\]
sin x dx=-dt
\[=-\int\limits t^2 dt +\int\limits t^4~dt=?\]