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#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Prove the following

$$\displaystyle \int^1_0 \frac{\log^3(1-x)}{x}\, dx=-\frac{\pi^4}{15}$$​

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#### chisigma

##### Well-known member
Prove the following

$$\displaystyle \int^1_0 \frac{\log^3(1-x)}{x}\, dx=-\frac{\pi^4}{15}$$​
With the substitution $\displaystyle 1-x=t$ the integral becomes...

$\displaystyle I= \int_{0}^{1} \frac{\ln^{3} t}{1-t}\ dt\ (1)$

... and according to the (5) in...

http://www.mathhelpboards.com/f49/integrals-natural-logarithm-5286/

... is...

$\displaystyle I = - 6\ \sum_{k=1}^{\infty} \frac{1}{k^{4}} = - \frac{\pi^{4}}{15}\ (2)$

Kind regards

$\chi$ $\sigma$

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Actually we can generalize for $$\displaystyle n\geq 2$$

$$\displaystyle \int^1_0 \frac{\log^{n-1}(1-x)}{x}\, dx =(-1)^{n-1} \int^{\infty}_0 \frac{x^{n-1}}{e^x-1}\,dx = (-1)^{n-1}\Gamma(n) \zeta(n)$$

For $$\displaystyle n = 4$$

$$\displaystyle \int^1_0 \frac{\log^{3}(1-x)}{x}\, dx = -\Gamma(4) \zeta(4)=-6\zeta(4) = -\frac{\pi^4}{15}$$