Interesting Integrals 12

Calculus Level 4

∫ 0 1 ( x ln ⁡ x ) 4 d x \large \int_0^1 (x\ln x)^4 \, dx

Find the closed form of the integral to 3 significant figures.


The answer is 0.00768.

Krishna Shankar by Krishna Shankar , 23, India

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1 solution

Chew-Seong Cheong
May 18, 2017

Relevant wiki: Gamma Function

I = ∫ 0 1 ( x ln ⁡ x ) 4 d x Let u = ln ⁡ x , e u = x , e u d u = d x = ∫ − ∞ 0 u 4 e 5 u d x Let − t = 5 u , − d t = 5 d u = 1 5 5 ∫ 0 ∞ t 5 − 1 e − t d t = Γ ( 5 ) 5 5 Γ ( ⋅ ) is the gamma function. = 4 ! 5 5 Note that Γ ( n ) = ( n − 1 ) ! ≈ 0.00768 \begin{aligned} I & = \int_0^1 (x\ln x)^4 dx & \small \color{#3D99F6} \text{Let }u = \ln x, \ e^u = x, \ e^u \ du = dx \\ & = \int_{-\infty}^0 u^4e^{5u} \ dx & \small \color{#3D99F6} \text{Let }-t = 5u, \ - dt = 5 \ du \\ & = \frac 1{5^5} \int^\infty_0 t^{{\color{#D61F06}5}-1}e^{-t} \ dt \\ & = \frac {\Gamma ({\color{#D61F06}5})}{5^5} & \small \color{#3D99F6} \Gamma(\cdot) \text{ is the gamma function.} \\ & = \frac {4!}{5^5} & \small \color{#3D99F6} \text{Note that } \Gamma(n) = (n-1)! \\ & \approx \boxed{0.00768} \end{aligned}

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