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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1
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1answer
84 views

Is changing variables the same as substitutions?

I have asked several on a similar matter. This time the question is tad different. $$\int_{\mathbb{R}} e^{-x^2} dx$$ We let $x=y \implies dx=dy$ $$\int_{\mathbb{R}} e^{-(x^2 + y^2)} dxdy$$ But ...
-3
votes
5answers
124 views

What allows change of variables? [duplicate]

In school, especscially, one is not taught "why" we can change variables, dummy variables in integration. $$\int_{a}^{b} f(x) dx$$ We can change the $x$ variable to $y$ for example. The idea is ...
2
votes
1answer
59 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
0
votes
0answers
28 views

Flux of a vector field through a defined area

I have to calculate the flux of the vector field $\vec A(\vec r) = \begin{pmatrix} e^{x^2+y^2} (x^2+y^2) \\ xy^2 ln(1+x^2) \\ e^{z^2} (z^2-2) \end{pmatrix} $ through the area defined by an ...
4
votes
1answer
61 views

Infinite Riemann sum of $x^3$

Write $$\int_{1}^{3} x^3 dx$$ as a riemann sum. Here is what I thought: $$\Delta(x) = \frac{2}{n}$$ $$f(x) = (\Delta(x)k)^3 = \left(\frac{2k}{n}\right)^3$$ $$I = \int_{1}^{3} x^3dx = \lim_{n ...
5
votes
2answers
66 views

Prove that $ \sum_{k=1}^\infty {\ln(k) \over k^2} \le{2+3\ln2 \over 4} $

Prove that $$ \sum_{k=1}^\infty{\ln(k) \over k^2} \le{2+3\ln2 \over 4} $$ Start with $$ \sum_{k=1}^n {\ln(k) \over k^2} \le \int_1^n {\ln(x) \over x^2}\,dx + f(1) $$ where $$ f(x) = {\ln(x) ...
2
votes
1answer
37 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
0
votes
1answer
55 views

Changing variables in Greens theorem

So we where learning yesterday about greens theorem and my teacher solved this integral for us $$ \oint{(x^3-2y+{x^2}Sin({x^3}+{y^3}))dx} + (2xy+{y^2}Sin({x^3}+{y^3}))dy $$ $$ \partial D: ...
0
votes
2answers
55 views

How to compute $I_n=\int^{+\infty}_{0}x^ne^{-x}dx,\ n \geq 0$? [closed]

I want to compute the following integral: $$I_n=\int^{+\infty}_{0}x^ne^{-x}dx,\ n \geq 0$$ How to do it?
1
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0answers
32 views

solving double integrals with two variables

I have an exam tomorrow that involves solving integrals like $$ \oint{Q dy + P dx}$$ by converting them to $$ \int{ \bigg(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\bigg)dy ...
0
votes
3answers
85 views

Calculate $\int\!\!\int (3x+4y) \, dy\, dx $ over the triangle with vertices $(0,0)$, $(3,3)$, and $(0,5)$.

Calculate the double integral $$\iint_T (3x+4y) \text{ } dy\text{ } dx $$over the triangle, $T$, with vertices $(0, 0)$, $(3, 3)$, and $(0, 5)$. I just need help with working out the limits
1
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1answer
95 views

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods?

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods? In all of the examples I have seen, poles are simple and this helps to avoid ...
1
vote
2answers
75 views

$\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$

I need to evaluate $$\displaystyle\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$$ I think I'll need $\displaystyle\int^{\infty}_{-\infty} e^{-x^2} ...
-1
votes
2answers
120 views

Definition of Bound/Free Variables

You may have already seen that: $$\int_0^1 x \, dx = \int_0^1 y \, dy$$ But the formal reason why this is done is because $x$ is a bound variable correct? QUESTION: We are allowed to change ...
3
votes
2answers
199 views

Nice book on integrals [duplicate]

On this site I usually see very amazing techniques to solve integrals; contour integrals, differentiating under the integral sign, transforming the integral into a series and son on and so forth. ...
0
votes
1answer
35 views

How can I evaluate this integral? - measure theory

Let $u \in L^{p}(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ and $2\leq p <\infty$. Let $u_{+}$ the positive part of $u$. I am trying to show that to show that $\int_{ \{ u \leq 0 ...
6
votes
1answer
124 views

Evaluate $\int \frac{\mathrm dx}{1+\cos^2 x}$

$$\int \frac{1}{1+\cos ^2x} \,\mathrm dx$$ I have to integrate the expression above: I tried with substitutions $\cos x=t$ and $1+(\cos x)^2=t$, but those didn't work, and I couldn't find any useful ...
1
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2answers
59 views

while finding PDF of $W=X+Y$ from given Joint PDF $f_{X,Y}(x,y)$ How to find the limits of integral?

RV $X$ and $Y$ have joint PDF: $$f_{X,Y}(x,y)= \begin{cases} 8xy & 0\le y \le x \le1 \\ 0 & \text{otherwise} \end{cases}$$ Find PDF of W=X+Y I know that I need to use : ...
7
votes
0answers
128 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
2
votes
0answers
38 views

Intuitive explanation why in some contexts logarithm shifted by Euler-Mascheroni constant is more natural

Natural logarithm is defined as inverse function to exponent. This way defined it has the value of $0$ in $x=1$. But if we define natural integral the following way ...
2
votes
2answers
34 views

Complex integral $\oint_L \frac{\cos^2{z}}{z^2}dz$

Compute $$ \int_L \frac{\cos^2 z}{z^2}\,dz$$ where $L$ is the closed loop that goes counterclockwise around the square with vertices $-1$, $-i$, $1$ and $i$. I was trying to compute this ...
1
vote
0answers
53 views

Almost everywhere convergence of convolution with mollifiers

I read that for $j\in L^1({\bf R}^n)$ with $\|j\|_1=1$ and $f\in L^1_{\rm loc}({\bf R}^n)$ the mollifiers $j_\epsilon(x):=\epsilon^{-n}j(x/\epsilon)$ exhibit $j_\epsilon\ast f\in L^1({\bf R}^n)$ and ...
1
vote
1answer
23 views

How to solve this integral with trigonometric functions?

How can I compute this integral manually? $\int_{1}^{t} sin2(t-\tau) cos2\tau d\tau$ I've tried some substitutions, trigonometric manipulations, but still cannot reach a reasonable next step. Any ...
0
votes
2answers
63 views

Winding Numbers of unit circle under $f(z)=\frac{z}{e^z-1}$

I've been working on the following question, and am stumped by part c. I know how to find the winding number if I have the graph, and I tried using the special case of the Cauchy Integral formula ...
2
votes
2answers
101 views

Definite integration involving square root function

How to integrate this definite integral: $$\int_{0}^{\pi/2} \big(\sqrt{\cos x}+ \sqrt{\cot x}\,\big)\,\mathrm dx$$
1
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3answers
58 views

Form multi-dimensional to one-dimensional: What's the general form of the integral?

From the indetities below $$ \int_0^{\infty}dn_1f(n_1)=\int_0^{\infty}f(n)dn\\ \int_0^{\infty}dn_1\int_0^{\infty}dn_2f(n_1+n_2)=\int_0^{\infty}nf(n)dn\\ ...
0
votes
0answers
72 views

convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral... \begin{equation} f_{Z}(z)=\int^{\infty}_{0}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = X_1 + X_2$ of two ...
4
votes
2answers
92 views

Show three ways that $f(z)=\frac{\overline{z}}{z-1}$ is not analytic

I need to show the complex function $$f(z)=\frac{\overline{z}}{z-1}$$ is not analytic in three ways; using Cauchy's equations, geometrically, and by integrating over the circle of radius 2. I used ...
0
votes
3answers
62 views

Integral inequality for a continuous and decreasing function on an interval

Let $f$ be a continuous and decreasing function on $[0,1]$. Show that $$\int_{0}^{\frac12}f(x)\,\mathrm dx \geq \frac12 \int_0^1 f(x)\,\mathrm dx$$
3
votes
3answers
125 views

Convergence of $\int_2^\infty \frac{dx}{x^2 \cdot (ln(x))^{\alpha}}$?

For which values of $\alpha > 0$ is the following improper integral convergent ?: $$\int_{2}^{\infty}{{\rm d}x \over x^{2}\ \ln^{\alpha}\left(\, x\,\right)}$$ I tried to solve this problem by ...
1
vote
2answers
102 views

For which values of $p \geq 0$ does the integral $\int_0^{\infty} \frac{dx}{x^{p}+x^{-p}}$ converge?

For which values of $p \geq 0$ does the integral $$\int_0^{\infty} \frac{dx}{x^{p}+x^{-p}}$$ converge? I tried applying the p-test, but I could not get the integrand into a suitable form. So what ...
0
votes
2answers
76 views

Fundamental Theorem of Calculus of a definite integral

$$\frac{\mathrm{d}}{\mathrm{d}x} \int_{2x}^{3x+1}\! \sin\left(t^4\right)\, \mathrm{d}t$$ could you just use the Fundamental Theorem of Calculus to get $$\sin\left(t^4\right) \bigg|_{2x}^{3x+1}$$ ...
5
votes
1answer
87 views

Mollifiers: Integral Bound

Desirable is an example such that: $$\varphi_0\in\mathcal{C}^\infty_0:\quad\int_0^\infty\left|\varphi_0^{(n)}(s)\right|\mathrm{d}s\leq2^n$$ (It should not exist as one would obtain entire elements for ...
0
votes
0answers
38 views

Is there a closed form polynomial for this integral recursion?

While working on some statistical problems, I startet playing with integral recursions of the type $$p_{n+1}(x)=\int_a ^x \mathrm{d} y\; q(y)p_n(xy)$$ Here $q(y)$ and $p_0(x)$ are given polynomials, ...
4
votes
2answers
148 views

a way to integrate: $\int (\sqrt{x} +3)/(2+ x^ \frac{1}{3}) dx$

Im looking for a way to integrate: $$ \int \frac{ \sqrt{x} +3}{2+ x^ \frac{1}{3}} dx $$ that would make it efficient and not too difficult... Any suggestions?
1
vote
1answer
65 views

Mollifiers: Integral Convergence

Why do these integrals converge: $$\varphi\in\mathcal{C}_0^\infty:\quad\frac{1}{\tau}\int_0^\tau\varphi(s)\mathrm{d}s\to\varphi(0)\quad(\tau\geq0)$$ I tried to figure it out via substituting: ...
2
votes
2answers
107 views

Lebesgue integral and absolute value

I wonder why we say that $f$ is integrable iff $\int|f|\,d\mu$ is finite? Why we use absolute value? Won't it be enough to have that $\int f\, d\mu$ is finite to call $f$ integrable? Are there ...
3
votes
2answers
73 views

Trigonometric Integration with Negative Exponents

How do you integrate $\csc^4 x/\cot^2 x$? I know that this is the same as $\csc^4 x \cot^{-2} x$ and when you use techniques in trig integrals you end up with $$\int \csc^2 x \csc^2 x \cot^{-2} x \,dx ...
4
votes
5answers
169 views

Integration question: $\int \sqrt{x^2-4}\, dx $

I am looking for a way to integrate $$\int \sqrt{x^2-4}\ dx $$ using trigonometric substitutions. All my attempts so far lead to complicated solutions that were uncomputable.
0
votes
2answers
79 views

Convincing argument that changing variables is justified

I for some reason cannot convince myself that changing dummy variables is justified. Suppose: $$I = \int_{0}^{1} x dx$$ I cannot convince myself that it is fine to change this dummy variable to ...
1
vote
3answers
78 views

Finding Volume of the Solid--washer method

The region between the graphs of $ y=x^2$ and $y=3x$ is rotated around the line $x=3$ What is the volume of the resulting solid? I drew the picture, and I saw that I should be using the washer ...
-1
votes
2answers
96 views

Replacing Variables in Integration [duplicate]

I have posted questions about this, but they werent clear, here is my actual misunderstanding. $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I dont understand, we say: $$I = \int_{-\infty}^{\infty} ...
2
votes
0answers
58 views

Find the length of the parametric curve (Difficult)

Find the length of the parametric curve $$x = t$$ $$y = f(t)$$ $$f(t) = \int_0^t {s \over (s^2-1)} \ \mathrm{d}s$$ $$0\leq t \leq 1/2$$ First I create the $x'$and $y'$ Then put it into the ...
0
votes
1answer
52 views

What does coefficient before Forier integral and integration limits depends on?

I've read a couple of sources on Fourier transform. All of them give different coefficients and integration limits. Wikipedia: 1, -infinity, +infinity. Russian Wikipedia: 1/sqrt(2*pi), -infinity, ...
0
votes
1answer
60 views

L-p space: p-norm proof

Can somebody put me in the right direction to prove that: $\lim_{p \to 1} \lVert f \rVert_{p}^p=\lVert f \rVert_{1}$ ? Maybe this will be a beginning: If $f \in$ $\mathcal{L}^1(\mu)\cap ...
3
votes
3answers
255 views

Proving an integral $\int \sqrt{a^2 - u^2} \, du$

How can I prove this?? Any hint or help would be great! Thanks :) $$\int \sqrt{a^2-u^2} \mathrm{d}u = \frac{u}{2} \sqrt{a^2-u^2} +\frac{a^2}{2} \sin^{-1}(\frac{u}{a}) + C$$
2
votes
2answers
614 views

integration by parts $ \int xe^{-2x} dx$

Can you guys help me integrate $ \int xe^{-2x} dx$ using integration by parts? So far I got an answer using this $$u = x \qquad dv = e^{-2x}dx \\ du = dx \qquad v = \frac{-e^{-2x}}{2} $$ so that ...
1
vote
1answer
35 views

How to solve initial value problem (problem written below)?

I usually know how to solve an initial value problem (move dx over to the side with the x variables and move y over to the dy side, then integrate both sides and solve), but this problem confused me. ...
4
votes
2answers
140 views

Approximation for elliptic integral of second kind

My (physics) book gives the following approximation: $\int_{-\pi/2}^{\pi/2} \sqrt{1-(1-a^2) \sin(k)^2} dk \approx 2 + (a_1 - b_1 \ln a^2) a^2 + O(a^2 \ln a^2)$ where a1 and b1 are "(unspecified) ...
5
votes
2answers
143 views

Stuck on integrating $\int x/(1-x)dx$

My attempt: Let $u = 1-x$ , $du = -dx$ , $x = 1-u$, so: \begin{align*} \int \frac{x}{1-x}\, dx &= - \int \frac{1-u}u\, du \\ &= - \left( \int \frac 1 u\, du - \int 1 \, du \right) \\ &= ...