Zeta + ln = Fun!

Calculus Level 5

∫ 0 1 x ( ln ⁡ x ) 2 2 ( 1 − x ) 2 d x = ζ ( p ) − ζ ( q ) \displaystyle \large \int_{0}^{1} \dfrac{x (\ln x)^2}{2(1-x)^{2}} dx = \zeta (p) -\zeta (q)

Find the value of p + q p+q .

Notation : ζ ( x ) \zeta (x) denotes the Riemann zeta function .


The answer is 5.

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Harsh Shrivastava by Harsh Shrivastava , 20, India

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1 solution

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Nice Handwriting

Department 8 - 5 years, 4 months ago

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Lol my handwriting is worst in my class.

Harsh Shrivastava - 5 years, 4 months ago

You could've used the relation of polygamma function with riemann zeta function also. See the wiki I've provided it there.

Aditya Kumar - 5 years, 4 months ago

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Integrating using Feynman's trick is fun.

BTW thanks for enlightening me with these cool relations :)

Harsh Shrivastava - 5 years, 4 months ago

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Yes Feynnman is a good method. But it becomes nasty at times.

Aditya Kumar - 5 years, 4 months ago

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