integration using substitution - question

Question

Transform this antiderivative problem using the substitution u = 5x + 4.

∫1/(5x + 4)3 dx

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

____________

Answer 1

u= 5x+4

du =5dx

dx =du/5

integral of dx/(5x+4)^3

=integral of du/5u^3

=(1/5)integral of (u^-3)du

=(1/5)(u^-2)/-2 + c where c = an unknown constant

= -1/10u^2 +c = the antiderivative in terms of u

= -1/10(5x+4)^2 + c = the antiderivative in terms of x

check the answer by taking the derivative

[-1/10u^2]'

=[-(1/10)u^-2]'

=(-2)(-1/10)u^-3

= 1/5u^3

or

[-1/10(5x+4)2]'

=[(-1/10)(5x+4)^-2]'

=-2(-1/10)(5)(5x+4)^-3

= (10/10)(1/(5x+4)^3

= 1/(5x+4)^3