# Evaluating definite integral by substitution

1. Oct 5, 2012

### dawn_pingpong

1. The problem statement, all variables and given/known data
By using substitution $$u=\frac{1}{t}$$, or otherwise, show that

$\int^∞_1 \frac{t^5}{(1+t^3)^3}dt=\int^1_0 \frac{u^2}{(1+u^3)^3}du$

2. Relevant equations

3. The attempt at a solution

Well, the reverse can also be done (making t to u). However, I don't know how to change the premise of the integral (from (∞,1) to (1,0). Thank you! I can integrate the integral after that.

2. Oct 5, 2012

### SammyS

Staff Emeritus
$\displaystyle \lim_{t\,\to\ \ \infty}\frac{1}{t}\ \ =\ \ \underline{\ \ ?\ \ }$

3. Oct 5, 2012

### dawn_pingpong

0?

so the integral is just $0- \int\left[\frac{(1/t)^2}{t^2(1+(1/t)^3}\right]^1$?(sorry don't really know how I'm supposed to write this...)

and at 0 the integral is 0. but it seems there is something wrong with the sign, because it is negative...

Last edited: Oct 5, 2012
4. Oct 5, 2012

### SammyS

Staff Emeritus
$\displaystyle \int_a^b f(x)\,dx=- \int_b^a f(x)\,dx$

5. Oct 6, 2012

### dawn_pingpong

oh right! thanks!