0
votes
1answer
54 views

Integral of the exponential function

I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.
0
votes
0answers
33 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right)$. Can this be found analytically?
1
vote
1answer
50 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
0
votes
1answer
29 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
79 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 ...
4
votes
1answer
150 views

Multiple integral over a disc

I would need some help on this integration problem: $$I=\int_0^{2\pi}\int_0^{R}\int_0^{2\pi}\int_0^{R}\exp(-a\ r_{12}) \ r_1 \ r_2 ...
3
votes
1answer
54 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.
3
votes
2answers
67 views

Calculation of integral $\int\exp \left(-\alpha \sin^2 \left(\frac{x}{2} \right) \right) dx$

Given $\alpha$ is a constant. How to calculate the following integral? \begin{equation} \int \exp \bigg(-\alpha \sin^2 \bigg(\frac{x}{2} \bigg) \bigg) dx \end{equation} Thanks for your answer.
1
vote
1answer
100 views

Double integrals of exponential functions

I need to find the double integral of $$e^{\frac{x}{y^2}}$$ bound by the $y\mbox{-axis}$, $x=y^2$, $y=1$, and $y=2$. The limits of integration were easy to find, but I am pretty confused about how to ...
2
votes
2answers
83 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
0
votes
1answer
16 views

Basic Integration + Root + Exponential issue

I have the following question : $\int ( 27e^{9x} + e^{12x} )^{1/3} dx $ However when I solved it I simplified it first to: $\int ( 27e^{9x} + e^{12x} )^{1/3} dx = \int \sqrt[3]{27e^{9x} + ...
4
votes
1answer
70 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
1
vote
2answers
34 views

Integrating two exponentials produces a cosine integral? Can somebody explain?

I discovered the following conversation that I do not understand. It reads: $$\int_{-U_1}^0 {(\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1+\int_0^{U_1} {(-\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1 = ...
1
vote
1answer
39 views

What is the integral containing decaying exponential function?

I am trying to figure out properties of the following integral: $$p(t)=\int_{0}^{t} e^{\alpha(t-t')} f(t')dt', \hspace{1 cm} t>t'$$ I would google and read more info about this integral but I do ...
-1
votes
2answers
68 views

Integration of $g(x) = e^{f(x)}$ [closed]

Is there any way of simplifying this integral? $$ f(x) = \int e^{2x^3}\,dx $$
1
vote
1answer
37 views

How to evaluate the integral $\int_0^{\ln3} e^{x-e^x}\,\mathrm dx$?

How to evaluate the following definite integral? $$\int_0^{\ln3} e^{x-e^x}\,\mathrm dx.$$ Should I use some sort of U Substitution?
0
votes
1answer
30 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
1
vote
3answers
61 views

Can you help me to get the integration of $\frac{1}{x} \exp(-x)$?

I need the solution of integral like $$\int^\infty_a \frac{1}{x} e^{-x} \,dx.$$ Thank you
7
votes
2answers
92 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
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: ${ds \over s}=\mu dt$ ...
5
votes
3answers
109 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
0
votes
2answers
83 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [closed]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
1
vote
1answer
39 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
1
vote
1answer
36 views

derivative of a definite integral with base e

$$\frac{d}{dx} \int_3^{x^2} e^{t^3} dt$$ I can sorta figure out how to solve problems like this, if it was an indefinite integral...
2
votes
1answer
45 views

A definite integral

$$\int_0^1\sqrt{\left(3-3t^2\right)^2+\left(6t\right)^2}\,dt$$ I am trying to take this integral. I know the answer is 4. But I am having trouble taking the integral itself. I've tried foiling and ...
0
votes
1answer
32 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
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
103 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
243 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 ...
0
votes
4answers
90 views

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.
3
votes
1answer
152 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
1
vote
2answers
245 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 ...
3
votes
1answer
93 views

how to double integrate this exponential function

how can i integrate this $\int_0^∞\int_x^∞{\frac{e^{-y}}{y}}dy dx$ i am stuck here $\int{\frac{e^{-y}}{y}}dy$. I have tried some log substitutions.
0
votes
1answer
42 views

Integral of exponential with second degree exponent

I want to compute the integral $$\int_\mathbb{R}e^{-t\left(y-\dfrac{(at+x)i}{2t}\right)^2}dy$$ I know that $\int_\mathbb{R}e^{-ty^2}dy=\sqrt{\pi/t}$, but here there is an extra imaginary factor. What ...
8
votes
0answers
117 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) ...
-1
votes
1answer
49 views

Help with solving $\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$

Given $$\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$$ Can you please show the step(s) involved to reach this next line in the textbook: $$\left[-\frac 1s\exp\left(\frac sr e^{-rt} \right) ...
0
votes
0answers
109 views

Integral involving Modified Bessel function, exponential and power

I am trying to evaluate the following integral: $$ \int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx $$ Even though there are a lot of integrals involving the ...
1
vote
1answer
55 views

Calculate $10,000e^{-\int_2^{10}\left(0.05+0.01/(t+1)\right)\,dt}$

This equation is used as an example in a text book with a given answer of $\approx$ 6,617 I cannot get to this solution as somewhere along the way I must be making an error. If it is a problem with ...
1
vote
0answers
302 views

Exponential integral approximation

I have an equation that contain exponential integral of the form: $$ \begin{equation} E_k\left(\frac{a+b ~x}{c}\right) \end{equation} $$ Where $k\geq 0$ ($k=0,1,2,...$), $a$, $b$, and $c$ are ...
1
vote
0answers
60 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
7
votes
1answer
104 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? $$\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$$
0
votes
1answer
92 views

Integral of a function and a derivative with respect to the same variable

So, I have the following solution to a differential equation: $$ y(t) = \frac{k\int \! e^{-pt} \frac{\mathrm{d}x(t)}{\mathrm{d}t} \mathrm{d}t}{e^{-pt}} $$ But I am not sure if I can cancel out the $ ...
9
votes
2answers
195 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
6
votes
0answers
135 views

Need help with $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

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 ...
0
votes
1answer
550 views

Integrate Exponential function with Integration Variable e (de)

I have the following problem: How do i integrate this: $ \int^1_0 e^{x^2+3 } de $ ? Whats putting me off is the de, i don't know how to integrate over the e-function... According to Wolfram Alpha, ...
6
votes
2answers
137 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
0
votes
1answer
101 views

How to put exp function back in the integral?

Thanks to everybody in advance. I have this integral: $$ e^{-at} \int_{0}^{t} e^{au}f(u) du $$ where f(t) is something I can't integrate. Is there a way to put the exponential function back in the ...
1
vote
1answer
90 views

Can anyone help with solving this integral?

I would like to solve this indefinite integral. I was practicing figuring out integrals when this one stumped me. WolframAlpha seems to have calculated it using some numerical method. I feel like this ...
1
vote
1answer
52 views

Solution to Singular Integral

Does anybody know how to solve the following singular integral: $$ \int_0^\pi \frac{\text{e}^{-i\cdot a\cdot \cos(x)}}{\cos(x)-b}\,\text{d}x $$ with $x$, $a$ and $b$ being real-valued. Do integrals ...
0
votes
0answers
44 views

Integration of function

I need to integrate $ 0.095x^{-0.26}[1-(1+\frac{a}{x})^{-0.2}]$. However, I am not sure how to multiply the exponents and then integrate it. Am I missing something easy?