0-pi-2-ln-sinx-ln-cosx-dx-

Question Number 160358 by Ar Brandon last updated on 28/Nov/21

\int_{0}^{\frac{π}{2}} \ln (\sin x) \ln (\cos x) d x

Answered by amin96 last updated on 28/Nov/21

\sin (x) = t \frac{dt}{dx} = \cos (x) = \sqrt{1 - t^{2}}

Ω = \int_{0}^{1} \frac{\ln (t) \ln (\sqrt{1 - t^{2}})}{\sqrt{1 - t^{2}}} dt = \frac{1}{2} \int_{0}^{1} \frac{\ln (t) \ln (1 - t^{2})}{\sqrt{1 - t^{2}}} dt

t^{2} = x \frac{dx}{dt} = 2 t = 2 \sqrt{x}

Ω = \frac{1}{4} \int_{0}^{1} \frac{\ln (\sqrt{x}) \ln (1 - x)}{\sqrt{x} \sqrt{1 - x}} dx = \frac{1}{8} \int_{0}^{1} \frac{(\sqrt{1 - x}) \ln (x) \ln (1 - x)}{\sqrt{x} \times (1 - x)} dx =

= - \frac{1}{8} \underset{n = 1}{\sum^{\infty}} H_{n} \int_{0}^{1} x^{n - \frac{1}{2}} {(1 - x)}^{\frac{1}{2}} \ln (x) dx =

= - \frac{1}{8} Σ H_{n} (Σ \frac{{(- 1)}^{n + 1}}{n}) \int_{0}^{1} x^{n - \frac{1}{2}} {(1 - x)}^{\frac{1}{2}} {(x - 1)}^{n} dx =

= \frac{1}{8} \underset{n = 1}{\sum^{\infty}} H_{n} {\underset{n = 1}{\sum^{\infty}} \frac{1}{n} \int_{0}^{1} x^{n - \frac{1}{2}} {(1 - x)}^{n + \frac{1}{2}} dx} =

= \frac{1}{8} \underset{n = 1}{\sum^{\infty}} H_{n} {\underset{n = 1}{\sum^{\infty}} \frac{1}{n} \int_{0}^{1} x^{(n + \frac{1}{2}) - 1} {(1 - x)}^{(n + \frac{3}{2}) - 1} dx} =

= \frac{1}{8} \underset{n = 1}{\sum^{\infty}} H_{n} {\underset{n = 1}{\sum^{\infty}} \frac{1}{n} β (n + \frac{1}{2}; n + \frac{3}{2})}

Commented by Ar Brandon last updated on 28/Nov/21

I wish it is further simplified .

Commented by amin96 last updated on 28/Nov/21

do you know the answer?

Commented by Ar Brandon last updated on 29/Nov/21

Yeah

Commented by Ar Brandon last updated on 29/Nov/21

\frac{π}{2} \ln^{2} 2 - \frac{π^{3}}{48}

Commented by amin96 last updated on 29/Nov/21

\int_{0}^{\frac{π}{2}} xln (sinx) \ln (cosx) dx

The answer to this integral

Commented by Ar Brandon last updated on 29/Nov/21

\int_{0}^{\frac{π}{2}} x \ln (\sin x) \ln (\cos x) d x = \frac{π^{2}}{8} \ln^{2} 2 - \frac{π^{4}}{192}

\int_{0}^{\frac{π}{2}} x \ln (\sin x) \ln (\cos x) d x = \frac{π}{4} \int_{0}^{\frac{π}{2}} \ln (\sin x) \ln (\cos x) d x