# Tagged Questions

Questions on the evaluation of definite and indefinite integrals

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### What's a better way to integrate this?

$$\int \frac{1}{x^2 + z^2} dx$$ I tried substitution and also by parts. By parts is getting messy and I am not sure if I am getting the right answer. I am trying to figure out an easier way or the ...
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### Flux integrals, parameterization

let S be the cylinder x^2 + z^2 = 9 where -2 /ge y /le 2 parameterization: thi(u,v)= <3cosv, u, 3sinv> where -2 /ge y /le 2 and 0 /ge v /le 2pi (thi is the symbol of I with the circle in the ...
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### Evaluating a line integral directly

$F(x,y) = \dfrac{1}{x^2+ y^2}\langle -y,x\rangle$, and let $R$ be a circle of radius $a$, centered at the origin. a) Why can't Green's theorem be used to evaluate $\int_R F \cdot ds$? b) ...
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### $\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$ , $\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$

Please help me integrate $$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$$ and $$\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$$ I've tried the standard $u = \tan \frac{x}{2}$ substitution but it looks ...
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### Integrating left to right versus right to left.

OK, I understand that when integration is done left to right with respect to x increasing left to right (dx is positive), that the answer is positive, and vice versa when integrating right to left. ...
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### Evaluating Complex Line Integrals

Calculate $\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$ and $\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$ when $\gamma$: $|z|=1$ is positively oriented. This is what I have tried to do, starting ...
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### $\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}$

I'm stuck on my last exercise. Could you help? $$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)} \ dx$$
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### What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?

I want to compute the following integral: $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$ with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$. ...
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### Integration of a rational function from +/- infinity

I am trying to calculate the integral $$\int_{-\infty}^{\infty}{\frac{a+x}{b^2 + (a+x)^2}\frac{1}{1+c(a-x)^2}}dx$$ where $\{a, b, c\}\in \mathbb{R}$. I have looked in a table of integrals for ...
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### $\int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx}$

I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct. $$\int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx}$$ For this one, ...
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### Integration involving roots

$$\int\frac{dx}{(1+x^\frac{1}{4})x^\frac{1}{2}}$$ This is my work: $$u^4=x$$ $$4u^3=dx$$ $$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u^3du}{(1+u)u^2}=-4(1+x^\frac{1}{4})^{-1}+2(1+x^\frac{1}{4})^{-2}+C$$ ...
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### The geometric interpretation [duplicate]

In the course of mathematical analysis, there was one problem that i excited to know more about it: What is the geometric interpretation of $$\int_a^b f(x)\,d(\alpha(x))$$ and $\alpha(x)$ is ...
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### Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$ I was told it could be calculated in a closed form.
I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...