# Tagged Questions

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### Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
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### Integral of $\int_{y_1}^{y_2} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$

Does the following integral have a closed form solution? $$\int_{y_1}^{y_2} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$$ $$0< y_1 < 1$$ $$0< y_2 < 1$$ Or is there an ...
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### Integral of exponential with $x(1-x)$ term

Does the following integral have a closed form solution? $$\int_{0}^{y} \exp\left(\,-\sqrt{\,x(1-x)\,}\,\right)\,{\rm d}x$$ Or must I settle with an approximation? Edit: Actual form of integral ...
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### Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
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### Normalizing a probability density function

I need to find a normalization term $N(\alpha,\beta)$ for the probability density function: $$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$ In other words, solve the following equation: ...
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### Contour integral $\int^\pi_{-\pi}(a-\cos\theta)^b\exp(c\cos\theta)d\theta$ assuming $a>1$, $b>0$, $c>0$

Under the condition $a>1$, $b>0$, $c>0$, is there any good function to express the following integral? $$\int^\pi_{-\pi}\left(a-\cos\theta\right)^b\exp\left(c\cos\theta\right)d\theta$$ I ...
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### Definite integration of a exponential function mixed with rational functions

Suppose $a>0$ , I am interested in a solution of the following definite integral: $$\int_{1}^{\infty}\frac{\exp({-az})}{z \sqrt{z^2-1}}{\,dz}$$ Thank you.
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### How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
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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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### Integrating exponential of exponential function: stuck at integration by parts

I want to integrate $$\int_{0}^{t}\exp\left\{{k_{1}\left ( 1-e^{-t/{k_{2}}} \right )}\right\}dt$$ First I substituted $u = 1-e^{-t/{k_{2}}}$ Thus I get ...
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### Integral $\frac{\sqrt{e}}{\sqrt{2\pi}}\int^\infty_{-\infty}{e^{-1/2(x-1)^2}dx}$ gives $\sqrt{e}$. How?

To calculate the expectation of $e^x$ for a standard normal distribution I eventually get, via exponential simplification: $$\frac{\sqrt{e}}{\sqrt{2\pi}}\int^\infty_{-\infty}{e^{-1/2(x-1)^2}dx}$$ ...
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### Definite integration of a high order exponential function mixed with rational function

I would like to solve the integral $$\int_{x>0}xe^{ax^m+bx^n}~dx,\qquad m>n>0$$
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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.): ...
I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...