# Tagged Questions

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### Integral equation solution

I have an integral equations of the form $\int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
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### Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
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### Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx$$ Here's what I have tried. We have to make a ...
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### Find a Harmonic conjugate $v(x,y)$ to $u(x,y)$.

Show that $u(x,y) = \frac{y^2}{x^3+y^3}$ in some domain and find the harmonic conjugate $v(x,y)$ to $u(x,y)$.
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### How to calculate $\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$ using Cauchy's integral theorem? [duplicate]

I need a hint. Where do I start if I want to calculate $$\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$$ using Cauchy's integral theorem?
### ${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$
I have a problem to solve integral $$I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}}$$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...