Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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2answers
29 views

Find $\iint_T{\frac{1}{2}xy\,dx\,dy}$ where T is the triangle with vertices at (1, 0), (3,2) and (5,0)

Please help me point out where I've gone wrong... I got the limits: $\int_{x-1}^2\int_1^5\frac{1}{2}xy\,dx\,dy$ Then changed the order of integration: $\int_1^5\int_{x-1}^2\frac{1}{2}xy\,dy\,dx$ ...
0
votes
1answer
28 views

Conceptual nature of reversing the limits of an integral

Let's say I integrate $$\int_5^0 x^2 \, dx$$ i.e.$-\frac{125}{3}$, what is physically happening with the summing of strips to produce a negative answer? Cheers Tom
0
votes
2answers
69 views

Integral problem 6 [on hold]

We know $$I_n=\int _0^1\:\frac{x^n}{x^2+1}dx\:,\:n\ge 0$$ and have show that: $$\lim _{n\to \infty }\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...+\left(-1\right)^{n-1}\cdot ...
2
votes
3answers
71 views

Integration problem for $x^4$ in the denominator

I'm stuck in solving the following problem. I'm getting no idea how to do this: $$Y = \int_{-k}^k \biggl(\frac{1}{N^2x^2-x^4}\biggr)\,dx$$ where $N$ is constant. Any leads will be highly ...
1
vote
4answers
58 views

Evaluate $\int_{0}^{1}(1-x)^ndx$ by expanding the bracket.

I'd like to get a hint on this exercise. I believe I'm somewhat close to the answer. I used the binomial theorem to get: $\displaystyle\int_{0}^{1}(1-x)^ndx = \int_{0}^{1}\sum_{k=0}^{n}{n\choose ...
2
votes
2answers
63 views

Computing $\int \tan(x)\,dx$ using Euler's formula

I attempted to integrate the $\tan(x)$ function by substituting the Euler's formula into $\tan x = \sin(x)/\cos(x)$. Integration resulted in the expression $\ln(2\cos(x))$, which is obviously not ...
0
votes
4answers
196 views

Integral inequality 5

How can I prove that: $$8\le \int _3^4\frac{x^2}{x-2}dx\le 9$$ My teacher advised me to find the asymptotes, why? what helps me if I find the asymptotes?
1
vote
1answer
38 views

Integrals with undefined points.

Suppose I have a function $$f(x) = \begin{cases} x, & \text{if $x$ }\in(0,1) \\[2ex] 0, & \text{if $x=0$} \end{cases}, f:[0,1]\to \Bbb{R}$$ Is $\int_0^{1}f$ defined? I feel like it should ...
1
vote
0answers
18 views

How to prove the inequality in Burgers equation and what is the relationship to energy dissipation

Consider Burgers equation: \begin{equation} u_t+uu_x-\epsilon u_{xx}=0 \quad , \quad u(x,0)=g(x) \quad , \quad u = 0 \mbox{ when } |x| \mbox{ large}, \end{equation} where ...
2
votes
1answer
30 views

How to solve this improper integral? [duplicate]

The problem is: If $f(x)\in C[0,+\infty)$, $\displaystyle\lim_{x\to+\infty}f(x)=k\in\mathbb R$, and $b>a>0$, prove: $$\int_{0}^{+\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-k]\ln(\frac ba)$$ My ...
1
vote
0answers
44 views

What is $\int (1-e^{-x})^n dx$?

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
0
votes
1answer
32 views

Finding present value using integrals

Find the present value A, to the nearest dollar, of a continuous annuity with: Annual rate of 6% (r=0.06) Time, T = 9 years If the rate of time t is at the rate ...
1
vote
0answers
25 views

Example of a function such that iterated integrals are equal

Is there any example of a function $f(x,y):[0,1]$x$[0,1]\to \mathbb R$ so that $\int_{0}^1\int_{0}^1f(x,y)dydx$ and $\int_{0}^1\int_{0}^1f(x,y)dxdy$ exists and are equal but $\int\int f(x,y)dydx$ does ...
0
votes
2answers
37 views

Can I move a constant function out of an integral?

I have the following integral: $$\int\exp(2\pi i \Omega t)\beta(\omega_0)d\Omega$$ while $\Omega$ is defined as $$\Omega = \omega - \omega_0$$. $\omega_0$ is handled as a constant value, while ...
1
vote
0answers
31 views

Analytically solving complicated integral involving logarithms.

I already asked a similar question a week ago and the comment I got helped me a lot with my progress, so that I now have new question to ask. I am stuck with solving a complicated integral and would ...
1
vote
1answer
32 views

Prove $\int^a_0 x(a^2-x^2)^{\nu-1} I_0(b x)dx= 2^{\nu-1}a^\nu b^{-\nu}\Gamma(\nu)I_\nu(a b)$

How can I prove the following equality? $$ \int^a_0 x\left(a^2-x^2\right)^{\nu-1} I_0\left(b x\right)dx= 2^{\nu-1}a^\nu b^{-\nu}\Gamma\left(\nu\right)I_\nu\left(a b\right), $$ under the ...
0
votes
2answers
25 views

Marginal Distribution: Integrate a variable out

Suppose we have given the joint density $f_{(X,Y)}(x,y)$ of two random variables $X, Y$, where $f_{(X,Y)}(x,y)=g(x,y) \mathbb{1}_{y > t}$. Now we want to compute the marginal density of $X$, ...
0
votes
1answer
19 views

Flux integral question

Evaluate the flux integral $\displaystyle \int \int_S {\bf F \cdot n} \ dS$ Where ${\bf F}(x,y,z) = z^2 {\bf k}$ where S is the part of the cone z^2 = x^2 + y^2 that lies between the planes z = 1 and ...
3
votes
1answer
57 views

Closed form for an almost-elliptic integral

Does $$\int_0^{2 \pi} \log\left(\frac{1}{2}[1+\sqrt{1-(a \sin\phi)^2}]\right) d\phi $$ have a closed form ? An approximation for small $a$ is $2E-\pi$, but it is the exact form that is needed ...
1
vote
1answer
36 views

Explaining why integral and summation can be swapped

I have the series $e^{-t^2}=\sum\limits_{k=0}^\infty \frac{(-1)^kt^{2k}}{k!}$ and I want to put it into the definition of $erf(z)$ to deduce the Maclaurin series for it. However, I have ...
-1
votes
2answers
17 views

Find volume of solid

Find volume of solid generated by revolving region in the first quadrant bounded by the coordinates axes, the curve y=e^-x and the line x=1, about y axis. Thanks
0
votes
1answer
42 views

Is the function integrable?

Is the function $$f(x) = \begin{cases} \frac{1}{10^n}, & \text{if $x$ }\in(2^{-(n+1)},2^{-n}) \\[2ex] 0, & \text{if $x=0$} \end{cases}, f:[0,1]\to \Bbb{R}$$ integrable? Find $\int_0^{1}f$. ...
0
votes
2answers
22 views

Finding an enclosed area using integrals

Find the area enclosed by $y=\frac 8{x^2}$, $y=x$, $x=8$ The answer says the area is equal to $27$. I tried dividing the area into $2$ (one of them a right angled triangle). I found the area of the ...
0
votes
1answer
20 views

$L^2$ integrability

Consider the function $f(x)=x^{-1/2}$ for $0<x<1$, and $0$ else. Let $\{r_n\}_{n=1}^{\infty}$ an enumeration of the rationals. Let $F(x)= \sum_{n=1}^{\infty}2^{-n}f(x-r_n)$. Prove that $F(x)$ is ...
2
votes
1answer
33 views

General solution of $\frac{dy}{dt}+\frac{t}{1+t^2}y=1-\frac{t^3}{1+t^4}y$

Find the general solution: $$\frac{dy}{dt}+\frac{t}{1+t^2}y=1-\frac{t^3}{1+t^4}y$$ We are currently learning about non homogeneous differential equations and the standard form we are given is: ...
0
votes
1answer
47 views

Expressing an integral in closed form

Is there a closed-form expression for this integral? $$\int \frac{\sin(Ax/2)}{A\sin(x/2)}\mathrm{d}x$$
5
votes
1answer
38 views

Find the general solution of the non-homogeneous equation.

Q. Find the general solution of the non-homogeneous equation. $$\frac{dy}{dt}+y=te^t$$ So here are my steps: Find $\mu (t)=e^t$ $y(t) e^t= \int e^t \times te^t$ So, I combined the terms and ...
0
votes
0answers
10 views

Surface Integrals

Find the surface are of the sphere x^2 + y^2 + z^2 = 3c^2 bounded by the function x^2 + y^2 = 2cz. Do it using spherical, cartesian, and polar coordinates. My answer was 4*(3)^(1/2)*c^3*pi. Thanks
0
votes
4answers
29 views

Revolving the area between two functions around the y-axis

The question comes from the 1991 AP Calculus test. Let $R$ be the region between the graphs of $y = 1 + sin(\pi x) $ and $y = x^2$ from $x = 0$ to $x = 1$. c) Set up, but do not integrate an ...
0
votes
2answers
42 views

Different results when integrating $1/(x \ln(x))$ partially/by substitution.

By substitution I get $ln(ln(x))$. Partially something completely different: $$\int \frac{1}{x \ln(x)} = \int \frac{1}{x} \frac{1}{\ln(x)} dx=\frac{\ln(x)}{\ln(x)} - \int -\frac{1}{x \ln(x) ^2} dx$$ ...
0
votes
1answer
19 views

Cartesian to Spherical Coordinate Conversion for Triple Integral

I have a question regarding what happens to the boundaries when converting a triple integral from Cartesian to Spherical Coordinates. Example ...
0
votes
2answers
64 views

Proving that $(e^x+1)^{1/3}$ has no elementary antiderivative

How should one prove that $$\int (e^x + 1)^{1/3}\, dx$$ is non-elementary? (In case that is really is)
0
votes
0answers
89 views

Using the “appropriate” formula

I am asked to solve $$\int_{C}\frac{1}{z+i}dz$$ where $C$ is parametrized $z(t) = 2+e^{it}$ for $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ by finding the antiderivative $F(z)$ of $f(z)$ and then ...
1
vote
0answers
27 views

Shell method resulted in a negative volume

The question is: Let R be the region bounded between the x-axis and the curve $y=x^2-4x$. Find the volume generated by rotating R about the y-axis. I used the shell method as the following ...
0
votes
1answer
35 views

Question About Continuity of an Antiderivative

Let $f(x)$ be a periodic function of period $T$ that is integrable over every finite interval. It is clear that $F(x)=\int_x^{x+T} f(x)dx$ is a constant. How can it be shown that $G(x)=\int_0^x ...
2
votes
2answers
71 views

Challenging integral

I am trying to find a close form representation for the following integral: $$ A(x;a,b,c)= \int_{0}^{x}\frac{\sin\left(a k+b k^{2}\right)+\sin\left(c k-b k^{2}\right)}{k}dk $$ for $0<x \ll ...
0
votes
1answer
22 views

Change of variables in higher dimension

The specific question I am interested in is the following: $\int_{x_2}^1 dx_1 \int_{x_3}^1 dx_2 \cdots \int_{x_{n+1}}^1dx_n \int_0^1 dx_{n+1}f_1(1-x_1)\cdots f_{n+1}(x_n-x_{n+1})$ and doing the ...
0
votes
1answer
48 views

How to solve one kind of integral problem analytically

I have to solve a kind of integral problem analytically : $$ \int_{-1}^{1}\frac{e^{-ik\sqrt{1-x}}}{\sqrt{1-x}}P_n(x)\text{d}x $$ where $P_n(x)$ is Legendre polynomial, $i$ is imaginary unit and $k$ is ...
0
votes
1answer
12 views

Derivation with respect to cumulative distribution function

I am trying to understand the how Baye, Kovenock and Vries 2008 calculate the derivative of E(w). F is a symmetric mixed-strategy equilibria with density f(x) on some subset (m,u) of the support of ...
0
votes
2answers
34 views

Paremetrising the Contour

I'm trying to paremetrise the Contour of a unit circle descibed anti clockwise. This is so I can integrate $$ \int_{|z| = 1} \frac{e^z}{4z^4} dz $$ Now normally $z(t)=e^{it}$ for $t\in [0,2\pi]$ is ...
0
votes
1answer
32 views

Is there any closed-form for this integral?

I have an expression $$F(\theta)=h(N)+2\sum_{l=1}^{N-1}h(N-l)\cos(\theta l)$$ I want to find the integral of the following $$\int_{\rm{lower}}^{\rm{upper}}(F(\theta))^2d\theta$$ here $\mathbf{h}$ ...
0
votes
0answers
22 views

How to find the integral, any closed-form?

I want to find the integral $$\int_{\rm{lower}}^{\rm{uper}}T\left({\rm{abs}}\left(\frac{1}{N}\sum_{n=-3}^{n=3}c_n\exp(-in\theta)\right)\right)^2d\theta$$ Here $c_n$, $T$ and $N$ are constants
3
votes
3answers
167 views

Integral of rational function with trigonometric functions

$$ \int \frac{dx}{(\sqrt{\cos x}+ \sqrt{\sin x})^4} $$ I saw this problem online and it looked like an interesting/difficult problem to try and tackle. My attempt so far is to use tangent half-angle ...
4
votes
1answer
31 views

Easy question on the limits of an integral

So I would like to ask how exactly do we determine what limits to take when integrating both Cartesian and parametric equations. So let's say we have a graph of $y=x^2$. If we wanted to take the area ...
2
votes
4answers
51 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
2
votes
2answers
22 views

What is the Coefficient Matrix of $T(p(t))=\int_0^t\int_0^yp(x)dxdy$ that maps $P_3\rightarrow P_5$?

The usual basis for $P_n$, of course, is given by $\left\{1,t,t^2,\cdots,t^n\right\}$. Why is the integrand a function of $x$? Does this matter for the purposes of constructing a change of basis ...
0
votes
1answer
31 views

Line integral confirmation and Geometric interpretation

I have $$\int_{C}(z - \bar{z})dz$$ where $C = \{z \; : \; |z-1| =2\}$ So I parametrize $C$ by letting $z = 2e^{it} + 1 = 2\cos(t)+ 1 + 2i\sin(t)$ and let $x = 2\cos(t)+1$ and $y = 2\sin(t)$, for $t ...
1
vote
1answer
30 views

Integrating the log-normal function

Compute $$F(t)=\int_0^t \frac{1}{\sqrt{2\pi}\sigma t} \exp\left[-\frac{1}{2}\left(\frac{\log t-\mu}{\sigma}\right)^2\right]\,dt; t>0$$ My Attempt: $u=\frac{1}{t}\Rightarrow du=-\frac{1}{t^2}dt$ ...
-3
votes
1answer
38 views

Length of an arc *calculus* [on hold]

I was stuck here because I don’t know what to do to integrate this Please help!!
0
votes
0answers
29 views

Improper Integral patterns [on hold]

Integration is one of those techniques that only gets better over time, as one does more problem, it becomes obvious when to use which integration technique. When to use substitution, by parts, etc... ...