16 questions linked to/from Integration using the Feynman Trick
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### Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

Prove that$$I=\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx=\frac{\pi}{3}\ln(2+\sqrt 3)+\frac43G$$ I've found this integral in my notebook and perhaps I encountered it before since it looks quite ...
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### Seeking methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|2 + \tan^2(x) \right| \:dx$

As part of going through a set of definite integrals that are solvable using the Feynman Trick, I am now solving the following: $$\int_{0}^{\frac{\pi}{2}} \ln\left|2 + \tan^2(x) \right| \:dx$$ I'...
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### What methods can be used to solve $\int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx$

I'm seeking methods to solve the following definite integral: $$I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx$$
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### Evaluate $\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx$

Prove that $$\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx=\frac {\pi \ln 2}{4}-\frac {G}{3}+\frac {2\pi}{3}\ln(2+\sqrt 3)$$ Where $G$ is the Catalan's constant. Actually I proved this using the ...
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### Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx$

After weeks of going back and forth I've been able to solve the following definite integral: $$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx$$ To solve this I employ ...