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### What's the common theme to look out for, while applying Feynman's Integral tricks? [duplicate]

I have seen Feynman's integral trick coming into use in many questions, but I don't really see a common way to recognize the format for it. Of course when a question comes with an explicit parameter, ...
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### Evaluating $\int_0^\pi x\frac{\sin{\frac{x}{2}} - \cos{\frac{x}{2}}}{\sqrt{\sin{x}}} dx$

How am I supposed to solve the following definite integral? $$\mathcal{I} = \int_0^\pi x \cdot \frac{\sin{\frac{x}{2}} - \cos{\frac{x}{2}}}{\sqrt{\sin{x}}} dx$$ This definite integral is solved if ...
• 379
554 views

### Evaulate $\int_{-\infty}^{\infty} \frac{\ln(x^2+1)}{x^4+x^2+1}dx$

I'm really stuck on this integral. I want to believe there is a closed form for it but I'm really unable to find it. WFA cannot find one either. I think it may be possible to use Feynman's Trick and ...
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### A difficult double integral $\int_{0}^{1}\int_{0}^{1}\frac{x\ln x \ln y }{1-xy}\frac{dxdy}{\ln(xy)}$

How to evaluate $$\int_{0}^{1}\int_{0}^{1}\frac{x\ln x\ln y}{1-xy}\frac{dxdy}{\ln(xy)} ?$$ Any ideas on how to even start with this integral? It seems impossible to me. There's a similar integral ...
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### Integral $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$

Prove that $$\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx =\frac{\pi}{4} \ln2$$ I tried to use King's rule and to scale by $2$ and then to add the integrals, to get product of ...
• 301
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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'...
### Integrating $\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2} dx$ using real methods
How to evaluate, without contour integration the following integral: $$I=\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2}\ dx\ ?$$ @Cody mentioned in this solution that I=\frac{\pi^{2}}{6}G+\frac{\pi^{3}}{...