- #1
Gregg
- 459
- 0
Homework Statement
show
[tex]\int _1^{\infty }\frac{1}{x^2}\text{Log}[x]dx=-\int_0^1 \text{Log}[x] \, dx [/tex]
similarly show
[tex] \int _0^{\infty }\frac{1}{x^2+1}\text{Log}[x]dx = 0 [/tex]
The Attempt at a Solution
For the first part a substitution 1/x works.
The second part I cannot do, I thought about
[tex] \int _0^{\infty }\frac{1}{x^2+1}\text{Log}[x]dx=\int _1^{\infty }\frac{1}{x^2+1}\text{Log}[x]dx+\int _0^1\frac{1}{x^2+1}\text{Log}[x]dx [/tex]
and then trying to maybe show
[tex] \int _1^{\infty }\frac{1}{x^2+1}\text{Log}[x]dx=-\int _0^1\frac{1}{x^2+1}\text{Log}[x]dx [/tex]
but for now I am not sure what to do.