Question

How do i find the integral(or antiderivative) of (3-x)(x^2-6x)^5

Answer 1

let u=the inside of nastier part.

Answer 2

Hello?

Answer 3

What do you mean?

Answer 4

u=x^2-6x
du=2x-6=2(x-3)

Answer 5

du=-2(3-x) dx

-du/2=(3-x)dx

Answer 6

What do I do with the 3-x though?

Answer 7

replace (3-x)dx with -1/2 du

Answer 8

Why do I replace it with -1/2 du?

Answer 9

you want to convert the integral from integral(f(x)dx) to integral(f(u)du)

if u=x^2-6x, then u'(x)=du/dx=2(x-3).
thus du=dx u'(x)=2(x-3)dx
thus -1/2*du=(3-x)dx

the original integral is integrate( (3-x)(x^2-6x)^5 dx)
=integrate( u^5 (3-x)dx)
=integrate ( u^5*(-1/2) du)
after integrating, convert back to x using the fact that u=x^2-6x

Answer 10

OH! okay I understand now. Thank you!!