Integral Integral10

Calculus Level pending

∫ 0 ∞ log ⁡ ( 1 x 15 + 1 ) d x = π 2 ( a + d b + c + b ) \int_0^{\infty } \log \left(\frac{1}{x^{15}}+1\right) \, dx=\pi \sqrt{2 \left(a+\sqrt{d \sqrt{b}+c}+\sqrt{b}\right)}

where a , b , c , d a,b,c,d are positive integers. Submit a + b + c + d a+b+c+d .


The answer is 30.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

Hint: ∫ 0 ∞ log ⁡ ( 1 x k + 1 ) d x = π csc ⁡ ( π k ) \int_0^{\infty } \log \left(\frac{1}{x^k}+1\right) \, dx=\pi \csc \left(\frac{\pi }{k}\right) when k > 1 k>1 (k real)

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