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## A trigonometric - logarithmic integral

Real & Complex Analysis, Calculus & Multivariate Calculus, Functional Analysis,
akotronis
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### A trigonometric - logarithmic integral

Evaluate $\displaystyle\int_{0}^{\pi/2}4\,\cos^2x\,\ln^2(\cos x)\,dx$.

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Grigorios Kostakos
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### Re: A trigonometric - logarithmic integral

\begin{align*}
\displaystyle\int_{0}^{\frac{\pi}{2}}{4\cos^2x\,\log^2(\cos x)\,dx}&\mathop{=\!=\!=\!=\!=\!=\!=\!=\!=\!=}\limits^{\begin{subarray}{c}
{t\,=\,\cos^2 x}\\
{-\frac{1}{\sqrt{t}\sqrt{1-t}}\,dt\,=\,dx}\\
\end{subarray}}\,4\int_{0}^{1}{\frac{t}{\sqrt{t}\sqrt{1-t}}\log^2\bigl({\sqrt{t}\,}\bigr)\,dt}\\
&=\int_{0}^{1}{\frac{\sqrt{t}}{\sqrt{1-t}}\log^2{t}\,dt}\,.\end{align*}
We have that \begin{align*}
I(m)&=\int_{0}^{1}{\frac{t^{m-\frac{1}{2}}}{\sqrt{1-t}}\,dt}=\int_{0}^{1}{t^{m+\frac{1}{2}-1}({1-t})^{\frac{1}{2}-1}\,dt}\\