Who's up to the challenge? 46

Calculus Level 5

$\large \int _{ 0 }^{ \infty }{ \dfrac { x\ln { x } }{ { e }^{ \sqrt { x } } } \, dx } =a-b\gamma$

If the equation above holds true for positive integers $a$ and $b$ , find $a+b$ .

Notation : $\gamma$ denotes the Euler-Mascheroni constant , $\gamma \approx 0.5772$ .

The answer is 68.

1 solution

Mark Hennings
May 4, 2016

With the change of variable $x = u^2$ we see that $\int_0^\infty \frac{x \ln x}{e^{\sqrt{x}}}\,dx \; = \; 4\int_0^\infty u^3 \ln u e^{-u}\,du \; = \; 4\Gamma'(4) \; = \; 44 - 24\gamma$ making the answer $44+24=\boxed{68}$ .

Who's up to the challenge? 46

The answer is 68.

1 solution

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