Question

How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

score 257 · Answer 1 · 2026-04-04 09:43:57

257

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How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

Published on 04 Apr 2026 - 9:43

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125

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117

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List Question

How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?

Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.

How to find the unit tangent vector of a curve in R^3

multiplying the integrands in an inequality of integrals with same limits

Closed form of integration

Proving smoothness for a sequence of functions.

Random variables in integrals, how to analyze?

derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound

Which type of Riemann Sum is the most accurate?

How to evaluate a Riemann (Darboux?) integral?

An elementary question on integrating

Approximate derivative in midpoint rule error with notation of Big O

Integral of ratio of polynomial

Integral with function calculation

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