# Tagged Questions

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### Help calculating this integral

Prove this for every $n>1$ (belongs to $\mathbb{N}$ ) $$\displaystyle \int_{0}^{1}\left( \frac{x^{2n+3} - x^{2n+1}}{1+x} \right) \, \mathrm{d}x =\frac{1}{2n+3} - \frac{1}{2n+2}$$ I don't see ...
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### Solving Integral that contain exponential and Power

I have an integral of this form: $$\int_0^\infty e^{-\frac{x}{a}-\frac{z^2}{bx}-\frac{z}{bx}}\left(\frac{c}{c+x+z}\right)^K~dx$$ where $K$ is a positive integer. $a$ , $b$ and $c$ are reals and ...
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### Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
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### Integrate $e^{-ax}$ and $xe^{-ax}$?

I'm making exercises about integration but I don't really get it. How do you solve these two integrals from 0 to +infinity? $\int Ae^{-ax}\,dx$ $\int Axe^{-ax}\,dx$ $A$ is a parameter.
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### Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals: $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$ for which I want to find ...
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### Evaluation of an integral involving hyperbolic sine and exponential

I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.): ...
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### Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...
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### What is the indefinite integral of $\int e^{\frac{1}{x^2 - a^2}} dx$?

I am looking for a solution to the following integral and finding it quite hard to find one ($|x| < a$): $$\int e^{\frac{1}{x^2 - a^2}} dx$$ I've tried to solve it with several substitutions, ...
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### Comparison test integral convergence

$$\int_0^{\infty} \frac{e^x}{x^x} \,\mathrm dx$$ How can I tell if this integral converges or not? I was thinking of using the comparison test, but I can't think of anything to compare it to. Could ...
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### Why the integral of $e^{-x}\;$ is $\;-e^{-x}$, and not $e^{-x}$?

I thought that the integral of $e^{x}$ is always $e^{x}$. Why does it change its sign to a negative when there is a negative exponent?
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### What is the fourier transform of this function?

With $$f(x) = \frac{1}{p} e^{-\pi x^2/p^2}$$ and $p>0$, I got an answer of $\displaystyle e^{-\pi p^2u^2}$. I just wanted to make sure I got the right answer. If I didn't, I will work through ...
### Is it true that $\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$
It seems to be true that: $$\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$$ For eg., this works with $\frac{dF}{dt}=\frac{1}{2} (\cos(\pi \ln{t})+1)$ But then there must be something ...