integration using substitution

Question

Transform this antiderivative problem using the substitution u = sin x.

∫sin4x cos x dx

Your final answer should be a new antiderivative problem written in terms of u.

__________

Answer 1

sin4x = 2sin2xcos2x

= 2(2sinxcosx)(1-2sin^2(x)

=4sinx(1-2sin^2(x))cosx

sin4xcosx = 4sinx(1-2sin^2(x))cosxcosx

=4sinx(1-2sin^2(x)sqr(1-sin^2(x)cosxdx

=4[(sinx -2sin^3(x))(sqr(1-sin^2(x)]cosxdx

= 4(u-2u^3)(sqr(1-u^2)du

integrate using integration tables or an integral calculator

= 4(1-u)(1-6u^2)(1+u)^(3/2)/15 + c

where c=an unknown constant

or solve with a 2nd subsitution

letting v= 1-u^2, dv=-2udu