1
vote
2answers
75 views

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 ...
7
votes
4answers
190 views

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 ...
4
votes
4answers
318 views

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 ...
0
votes
0answers
49 views

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 ...
0
votes
1answer
32 views

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: ...
1
vote
2answers
85 views

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 ...
3
votes
1answer
64 views

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.
1
vote
1answer
44 views

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: ${ds \over s}=\mu dt$ ...
3
votes
1answer
54 views

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 ...
2
votes
1answer
109 views

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 ...
5
votes
2answers
248 views

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 ...
1
vote
2answers
261 views

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 ...
0
votes
1answer
54 views

confusing intergration

given $$r=4e^{3\theta} \space \space \space \space dr/d\theta=(3*4*e^{3\theta})$$ $$l=\int \sqrt(4e^{3\theta})^{2}+(3*4*e^{3\theta})^{2} \rightarrow $$ why does the integral $$ ...
9
votes
0answers
129 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
2
votes
1answer
145 views

Integrating exponential of multiple exponentials

I have a integral term that looks similar to $\int_0^\infty\exp(-u-ae^{-c_1u}-be^{-c_2u})\,du$ where the constants $a,b,c_1,c_2>0$. For the case where $b=0$ I can use the answer from: Integrating ...
7
votes
1answer
108 views

Need your help with the integral $\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$.

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$$
2
votes
2answers
96 views

Evaluating $\int _{-1}^{e} \frac{1}{x}dx$

Here very easily by the Fundamental Theorem of Calculus $$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$ From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the ...
8
votes
1answer
193 views

Closed form of $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
3
votes
2answers
138 views

How do we calculate this exponential integral if we change limit from $\infty$ to $x_1$

We know the solutions of this integral (Bronstein-Semendijajev mathematics manual [page474]): \begin{align} \int\limits_{0}^{\infty}x^n \cdot e^{-ax^2}dx = ...
2
votes
4answers
149 views

Some exponential integrals - I need algebraical solution besides my graphical one

I have come across integrals of form: \begin{align} &\int\limits_{-\infty}^{+\infty} x\cdot e^{-ax^2} dx\\ &\int\limits_{-\infty}^{+\infty} x^2\cdot e^{-ax^2} dx\\ ...
2
votes
1answer
41 views

Parametric solution of a multiple integral

Let $b$ and $a_i, i = 1,2,3,..., n$ be positive real numbers such that $$ \int_{0}^{\infty}\int_{0}^{\infty}\ldots \int_{0}^{\infty} e^{-b x_1^{a_1}x_2^{a_2}\ldots x_1^{a_2}} dx_1 dx_2 \ldots dx_n ...
2
votes
1answer
161 views

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: $$\frac{\sqrt{e}}{\sqrt{2\pi}}\int^\infty_{-\infty}{e^{-1/2(x-1)^2}dx}$$ ...
1
vote
1answer
258 views

Definite integration of a high order exponential function mixed with rational function

I would like to solve the integral $$\int_{x>0}xe^{ax^m+bx^n}~dx,\qquad m>n>0$$
2
votes
1answer
145 views

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.): ...
1
vote
2answers
416 views

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...
1
vote
2answers
348 views

Change of variables for triple integrals

Question: Consider the one-to-one transformation $(u; v;w) \to (x; y; z)$ defined by the equations $u = x + y + z; uv = y + z; uvw = z;$ which maps the unit cube $U$ defined by $0 \lt u \lt 1, 0 < ...
5
votes
1answer
149 views

How to evaluate $\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$?

How can I integrate the following: $$\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$$ for $a,b>0$? Maple gives a compact result: $$\frac{1}{b} \tan^{-1}(c) - \frac{1}{ac^2} ...
5
votes
1answer
167 views

$\int_a^bx^\alpha \exp\left[-\left(\frac{x-\delta}{\sigma}\right)^\alpha\right]dx$ Closed-form?

I am trying to evaluate the following integral: $$\int_a^bx^\alpha \exp\left[-\left(\frac{x-\delta}{\sigma}\right)^\alpha\right]dx$$ where $0<a<b$, $\delta\leq a$ and $\alpha>0$. Is ...