Harmonic integral

Calculus Level 4

$\large \int_{-1/2}^0 H_ x \, dx = \dfrac ab \gamma - \dfrac cd \ln \pi$

If the equation above holds true for positive integers $a,b,c$ and $d$ with $\gcd(a,b) = \gcd(c,d) = 1$ , find $a+b+c+d$ .

Notations :

$H_n$ denote the generalized harmonic number .
$\gamma$ denotes the Euler-Mascheroni constant .

The answer is 6.

1 solution

Aareyan Manzoor
Mar 19, 2016

We use the digamma function

$\int H_x dx=\int \psi(x+1)+\gamma dx=\ln(\Gamma(x+1)) +\gamma x+C$ so, $\int_{-1/2}^0 H_x dx=\ln(\Gamma(1))-\left(\ln\left(\Gamma\left(\dfrac{1}{2}\right)\right)-\dfrac{1}{2}\gamma\right)\\=\dfrac{1}{2}\gamma-\dfrac{1}{2}\ln \pi$

Same way! +1

Nihar Mahajan - 5 years, 2 months ago

Harmonic integral

The answer is 6.

1 solution

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