3
votes
1answer
57 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 ...
-1
votes
0answers
18 views

How to integrate this function and decide lambda

I want to decide lambda for I also want to integrate the same function. Please help!
0
votes
2answers
76 views

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

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
26 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
33 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...
1
vote
1answer
43 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
27 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
51 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
91 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
211 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
67 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.
2
votes
1answer
123 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
129 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
66 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
39 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 ...
7
votes
0answers
95 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
43 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
92 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
46 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
229 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
54 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
87 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
85 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 $ ...
8
votes
2answers
184 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
119 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
509 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
131 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
99 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
87 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
50 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?
1
vote
3answers
189 views

How to solve the definite integral $\int_{-4}^{-2}e^{-x}\,dx$?

I'm trying to find the value of the integral $\int_{-4}^{-2}e^{-x}\,dx$ but I just couldn't solve it. Actually I found in a List of integrals that $\int e^x\,dx=e^x+C$ so I concluded: ...
0
votes
1answer
68 views

Matrix integral of absolute exponential item

If $A=(a_{ij})$ is an $n\times n$ symmetric positive matrix, is it possible to calculate the following matrix integral? $$\int_{0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |\mathrm dt,$$ where ...
3
votes
2answers
118 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
138 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
3answers
221 views

How to prove gaussian-like integral equation true.

Integrals is definitely not my strong point, and I'm having trouble proving that: $${\int_{-\infty}^\infty (e^{\large{\pi}n})^{-\large{x}^2} dx = {1\over\sqrt{n}}}$$ It has similarities to the ...
1
vote
2answers
54 views

Please help to solve this ODE with function coefficients

Is it possible to solve this ODE for $y$? According to wikipedia this falls in the category of a first order, linear, inhomogeneous ODE with function coefficients. But is there a more tractable ...
0
votes
2answers
75 views

Solving an integral of form $ F(x)=\int (2x-1)e^{2x}\ dx $

I have this integral in a worksheet please help me solve it $$ F(x)=\int (2x-1)e^{2x}\ dx $$
0
votes
0answers
168 views

integrate normal cdf

Is it possible to integrate the following equation. It is a product of a cumulative normal distribution used and an exponential function. I tried mathematica online but it fails. If it is not ...
1
vote
3answers
75 views

Evaluating $\int_{a}^{\infty} \tfrac{(t-a)^2}{C^2} \cdot \exp ( - \tfrac{(t-a)^2}{C^2} ) \cdot \exp(iwt) \ dt$

I am trying to evaluate the following integral: $$\int_{a}^{\infty} \tfrac{(t-a)^2}{C^2} \cdot \exp ( - \tfrac{(t-a)^2}{C^2} ) \cdot \exp(iwt) \ dt$$ I thought about using the substitution $\beta ...
5
votes
1answer
110 views

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 ...
2
votes
1answer
136 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
366 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
1answer
96 views

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, ...
2
votes
3answers
49 views

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 ...
2
votes
4answers
89 views

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?
1
vote
1answer
43 views

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 ...
1
vote
1answer
36 views

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 ...
3
votes
1answer
149 views

$\iiint e^{-x^2-2y^2-3z^2}dV$

I was given this question in class but I don't understand how to do it. Evaluate the triple integral in $\mathbb{R}^3$ of $\iiint e^{-x^2-2y^2-3z^2}dV$. The hint was to use the idea that $\int ...
6
votes
1answer
255 views

integral of the product of a trigonometric and an exponential function

Since tan has an odd power I would normally aim to sub $u=\sec(x)$, but I cant get rid of the $2^x$. $$\int 2^x \tan^9(x^2)\sec(x^2)dx$$ I also tried integrating by parts but it got more complicated. ...