Evaluating definite integral by substitution

[dawn_pingpong]

Homework Statement

By using substitution $$u=\frac{1}{t}$$, or otherwise, show that

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

Homework Equations

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.

 

[SammyS]

Homework Statement

By using substitution $$u=\frac{1}{t}$$, or otherwise, show that

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

Homework Equations

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.

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

 

[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...

 

[SammyS]

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

 

[dawn_pingpong]

oh right! thanks!