# Math Help - Prove the values of the improper integrals.

1. ## Prove the values of the improper integrals.

Prove that

$
\int_{0}^{\infty}x\frac{ ln(1+x^3)}{1+x^3}dx
= ln\ 3 \ \frac{\pi}{\sqrt{3}} + \left(\frac{\pi}{3}\right)^2$

$
\int_{0}^{\infty} \frac{ln(1+x^3)}{1+x^3} dx
= ln\ 3 \ \frac{\pi}{\sqrt{3}} - \left(\frac{\pi}{3}\right)^2$

EDIT: Yep, I know the first one is a duplicate of the improper integral I posted last time, but what about the second one?

2. Originally Posted by mathwizard
Prove that

$
\int_{0}^{\infty}x\frac{ ln(1+x^3)}{1+x^3}dx
= ln\ 3 \ \frac{\pi}{\sqrt{3}} + \left(\frac{\pi}{3}\right)^2$

$
\int_{0}^{\infty} \frac{ln(1+x^3)}{1+x^3} dx
= ln\ 3 \ \frac{\pi}{\sqrt{3}} - \left(\frac{\pi}{3}\right)^2$

EDIT: Yep, I know the first one is a duplicate of the improper integral I posted last time, but what about the second one?
Let $
J\left( \theta \right) = \int_0^\infty {\tfrac{{\ln \left( {1 + \theta ^3 \cdot x^3 } \right)}}
{{1 + x^3 }}dx}
$
differentiating by Leibniz rule: $
J'\left( \theta \right) = 3 \cdot \theta ^{2} \cdot \int_0^\infty {\tfrac{{x^3 }}
{{\left( {1 + \theta ^3\cdot x^3 } \right) \cdot \left( {1 + x^3 } \right)}}dx}
$

Remember that: $
\int_0^\infty {\tfrac{{dx}}
{{1 + x^3 }}} = \tfrac{{2 \cdot \pi }}
{{3 \cdot \sqrt 3 }}
$
, calculate the derivative and integrate (considering that $
J\left( 0 \right) = 0
$
) to get $J(1)$