Question

What did I do wrong? -Calc. Substitution-

Problem: [2-1<-- (this is from x=2 to x=1) x(x-1)^1/2 dX

My Answer and work:
Substitute U for x-1
du/dx = 1
du=dx

[2-1 x (u)^1/2
(x^2)/2 * (u^3/2)/(2/3) ]2-1
(1/2(2)^2 * (3/2)1^(3/2)) - (1/2(1)^2*(3/2)(0)^(3/2)
This comes out to 1.5 but the answers in the back of the book say it is 16/15. Can someone show me where I went wrong and get me to the right answer?

Answer 1

You left an x when you substituted. Even integrated it.
You did u=x-1. So x=u+1
If you do this, you end up with (u+1)*sqrt(u) which you can distribute

Answer 2

Ohh thanks, I didnt realize that you had to pay attention to that X. Will you finish the problem?

Answer 3

Sure.
(u+1)*sqrt(u)
Distribute
u^(3/2)+u^(1/2)
Integrate and you get:
(2/5)u^(5/2)+(2/3)u^(3/2)
We also change the limits of integration using u=x-1
So we are integrating from u=1 to u=0
Evaluating this we get
-16/15

If integrating from x=1 to x=2, this same procedure gives
16/15

Answer 4

thanks!