A calculus problem by Saswata Naha

Calculus Level 3

Evaluate ∫ e ∞ 1 x ln ⁡ x ln ⁡ x d x . \large \int_e^\infty\frac {1}{x \ln x\sqrt {\ln x}} \mathrm dx.

1 2 3 Can not be evaluated

Saswata Naha by Saswata Naha , 20, India

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1 solution

Chew-Seong Cheong
Apr 27, 2017

I = ∫ e ∞ 1 x ln ⁡ x ln ⁡ x d x Let u = ln ⁡ x , e u = x , e u d u = d x = ∫ 1 ∞ e u e u u u d u = ∫ 1 ∞ u − 3 2 d u = − 2 u − 1 2 ∣ 1 ∞ = 2 \begin{aligned} I & = \int_e^\infty \frac 1{x\ln x \sqrt{\ln x}}dx & \small \color{#3D99F6} \text{Let }u = \ln x, \ e^u = x, \ e^u \ du = dx \\ & = \int_1^\infty \frac {e^u}{e^u u \sqrt u}du \\ & = \int_1^\infty u^{-\frac 32} du \\ & = - 2 u^{-\frac 12} \bigg|_1^\infty \\ & = \boxed{2} \end{aligned}

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Nice substitution to solve it!

Peter van der Linden - 4 years, 1 month ago

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Glad that you like it.

Chew-Seong Cheong - 4 years, 1 month ago

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I used a similar substitution, but worked wit d u = 1 x d x du = \frac{1}{x} dx . It gives the same integral in the end of course.

Peter van der Linden - 4 years, 1 month ago

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