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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2
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1answer
32 views

On an generalized integral exercise: $ \int_{0}^{+\infty} \frac{dx}{\sqrt{x} | 1-x |^{\alpha}} $.

I am asked to determine for which $\alpha > 0$ does the following generalized integral converge: $$ \int_{0}^{+\infty} \frac{dx}{\sqrt{x} | 1-x |^{\alpha}} $$ I did the following $$ ...
2
votes
2answers
75 views

Evaluate $\int_a^b(x-a)^3(b-x)^4 dx $

Evaluate $\int_a^b(x-a)^3(b-x)^4 dx $ with $0<a<b$. I've tried the following substitutions : $ y = b - x$ Giving, $\int(b-a-y)^3(y)^4 dy $ Then, $ z = y / (b-a)$ Giving, ...
1
vote
1answer
39 views

Question on $ L^p $ spaces inequalities to prove limit exists

in my class on real analysis we are currently dealing with $ L^p $ spaces and I have been tackled with this problem from Folland's real analysis stating this: If $ f $ is absolutely continuous on ...
1
vote
3answers
56 views

Show that the improper integral converges

Let $f[0, \infty)\to\mathbb{R}$, differentiable, positive and $\lim_{x\to\infty} (\log f)'(x) = L < 0$. Prove that $\int_0^\infty f < \infty$. So $\lim_{x\to\infty} (\log f)'(x) = L$ ...
0
votes
1answer
50 views

Line integral using Green's Extended Theorem.

Let $C$ be parametrization $\mathbf{r}=\space5\cos(t) \mathbf{i}+\space4\sin(t) \mathbf{j}\space, t \in [0, 2\pi]$. Calculate $\oint_C \mathbf{F}\cdot d \mathbf{r}$ where $F$ is vector field ...
0
votes
0answers
23 views

What would the limits of integration be for this double integral?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy}dx \ dy$, where $A$ is the area defined by $x>0, \ y>0, \ x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x^{2}} \leq y \leq ...
1
vote
1answer
61 views

Evaluating $\int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}\,\mathrm{d}x $

I'm trying to figure out this integral but cannot figure out the right substitution $$ \int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}dx $$
0
votes
1answer
48 views

Changing the order of integration?

For example if I have: $$\int_0^1 \int_1^2 \int_0^3 f(x,y,z) \space dx \space dy \space dz $$ If I want to change the order of integration e.g. to $( dz \space dy \space dx )$ $$\int_0^3 \int_1^2 ...
0
votes
1answer
48 views

Find the formula $\int_0^1f(x)dx\approx A_0f(0)+A_1f(1)$ that is exact for all functions of the form $f(x)=ae^x+b\cos(\pi x/2)$. [closed]

Find the formula $\int_0^1f(x)dx\approx A_0f(0)+A_1f(1)$ that is exact for all functions of the form $f(x)=ae^x+b\cos(\pi x/2)$. I'm not sure how to go about this. Any solutions/hints are greatly ...
0
votes
1answer
20 views

Double integration (correction)

I need your help to verify my answer. This is an exercice who I have to make but the manual doesn't give the answer. I need to calculate the charge : $\sigma(x,y)=x+y+x^2+y^2$ on the disc : $x^2+y^2 ...
1
vote
2answers
52 views

Differentiate an exponential integral

Would you guide me differentiating this integral for $m$: $$\frac{\text{d}}{\text{d}m} \int_{x=-\infty}^m\int_{y=n}^{+\infty}\exp\left(-\left[\left(\frac{x-a}{b}\right)^2 - ...
0
votes
2answers
29 views

First order differential equation - area.

a) Solve the differential equation: $$(x+1)\frac{dy}{dx}-3y=(x+1)^4$$ given that $y=16$ and $x=1$, expressing the answer in the form of $y=f(x)$. b) Hence find the area enclosed by the graphs ...
1
vote
5answers
54 views

Evaluate the integral $\iint\limits_R \sqrt{1-x^2} {d}A$

If $R = \{(x, y) | -1 \leqslant x \leqslant 1, -2 \leqslant y \leqslant 2 \}$, evaluate the integral $$ \iint\limits_R \sqrt{1-x^2} {d}A $$ The author of the book ('Multivariable Calculus' by James ...
3
votes
2answers
25 views

Finding the inverse laplace of this function: $ F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $ F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$ F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
3
votes
2answers
61 views

I need integrate this $\int_{} \frac{1}{\sqrt{1-z^2}-z} dz$

I don't know how to integrate this $\int_{} \frac{1}{\sqrt{1-z^2}-z} dz$ I tried with suspstitution $ t=\sqrt{1-z^2}-z $ but it doesn't work. Please help!
0
votes
0answers
33 views

Fourier Transform of a kernel

Let $\omega \in \mathcal{S}(\mathbb{R}^{2})$ and define $u(x) = \int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x - y|^{2}}\omega(y)dy$, where $(x-y)^{\perp} = \begin{bmatrix}x_{2} - y_{2}\\-x_{1} + ...
1
vote
2answers
75 views

Proof that Beta-function $B(m,n)$ = $\frac{n-1}{m}B(m+1,n-1)$?

When m and n are positive integers. It probably has to do with the incomplete Beta-function $B_{sin^2(x)}(m,n)$.
0
votes
1answer
42 views

How do I calculate the Beta-function $B(m,n) = 2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt$

The Beta-Function $$B(m,n) =2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt \tag{a}$$ is equal to $$\frac{n-1}{m}B(m-1,n+1) \tag{b}.$$ How do I go from (a) to (b)? (I tried with ...
1
vote
1answer
41 views

What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$?

What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$? AFAIK : $f(x)$ is odd function $(x-\pi)^2$ should be even because of square, and it's odd because of $(\sin x)$. Can you explain in ...
0
votes
1answer
28 views

Curve integral, what am I doing?

Suppose $f(x,y)=x^2y^2$ and path defined $\alpha(t)=(\cos t, \sin t)$, $0\leq t\leq 4\pi$ Going to find $\int_{\alpha}f ds$, but not completely sure in the final part. Ok, first $x(t)=\cos t$ and ...
3
votes
2answers
72 views

Evaluate $\int_0^\pi \frac{x}{1+\sin\alpha\sin x} dx$

$$\int_0^\pi \frac{x}{1+\sin \alpha \sin x} dx$$ I need some Hints about how to begin with the problem, because I can't think of anything
0
votes
1answer
30 views

General technique to check the convergence of an improper integral?

Which of these integrals converge ? I am confused about how to check for the convergence when the functions are more complex inside the integral. My attempt: in option C : integrating gives -2 and ...
0
votes
1answer
36 views

$\int_0^4|x^3-2x^2-x+2|dx$ I didn't get the correct end result and I'm searching after the bad step in my solution.

Well, as I calculated, that $(x^3-2x^2-x+2)$ is equal to $(x-2)(x-1)(x+1)$, so I have to check where is $(x^3-2x^2-x+2)dx\ge0$ $\implies$ $(x-2)(x-1)(x+1)\ge0$ and I get the intervals of ...
3
votes
1answer
30 views

How to Prove $\oint ({\mathcal{\hat{r}}} \cdot \vec r') \mathrm{ d\vec l'} = -{\mathcal{\hat{r}}} \times \int \mathrm{d\vec a'}$?

$$\oint ({\mathcal{\hat{r}}} \cdot \vec r') \mathrm{ d\vec l'} = -{\mathcal{\hat{r}}} \times \int \mathrm{d\vec a'}$$ Here the integration in the LHS is around a certain loop and the $d\vec a'$ ...
0
votes
0answers
43 views

Partial volume of inclined horizontal cylinderical tank with frustum(Truncated Cone) ends.

Partial volume of inclined horizontal cylindrical tank with frustum(Truncated Cone) ends. Inner Length - 16225mm Inner Diameter - 4000mm Trunc. Cone Dia ...
3
votes
0answers
65 views

How to evaluate: $\int_0^1x^{n-1}(1-x)^{n+1}dx$

How can I evaluate the following integral? ($n \in R$, $n>0$) $$\int_0^1x^{n-1}(1-x)^{n+1}dx$$ I was solving the following problem (as practice) in school: Prove that the sum of $n+1$ terms of ...
2
votes
2answers
33 views

Integrating a second derivative

Admit that $f$ has a second derivative find the integer $m$. $$m\int_{0}^{1}xf''(2x)dx = \int_0^2xf''(x)dx$$ So I took $2x=u$ where $du/dx=2$ and I plugged in the integral getting ...
1
vote
2answers
61 views

Integrating $\int_{-\infty}^{\infty} e^{-2b x^2} \, dx$ [duplicate]

I need help in integrating this integral: $$\int_{-\infty}^{\infty} e^{-2b x^2} \, dx$$ Knowing the value of $\int_{-\infty}^{\infty} e^{-x^2} \, dx$, and the integral for $e^{bx}$, I'm guessing ...
0
votes
2answers
29 views

The continuity of a function that can't be expressed in terms of elementary functions

The function $F(x) : = \int \limits _x^{x^2} \dfrac{\sin t}{t} dt$ is differentiable on $(1, \infty)$. How can one go about showing that this is true? Since you can't express this integral in ...
2
votes
3answers
80 views

Show that if $f$ is continuous on [0,1], then: $\int_0^\frac\pi 2 f(\sin x)dx=\int_0^\frac\pi 2 f(\cos x)dx= \frac12\int_0^\pi f(\sin x)dx$

Also part of the problem: Show that: $$\int_0^{n\pi} f(\cos^2 x)dx=n\int_0^{\pi} f(\cos^2 x)dx$$ It may be important to note that I have taken a few semester of calculus prior to this, and that this ...
0
votes
1answer
29 views

Proving that a function is L1

Suppose $f \in L^1([0,b])$ and $g(x)=\int_x^b{\frac{f(t)}{t}dt}$ , prove that $g\in L^1([0,b])$ and $\int_{0}^{b} g(x) dx = \int_{0}^{b} f(t) dt$. Assume we are not allowed to use integration by ...
3
votes
0answers
46 views

What is the number of zeros of antiderivatives of $(x-1)(x-2)^2(x-3)^3(x-4)^4$?

For each $x \in \mathbb{R}$, let $f(x) = (x-1)(x-2)^2(x-3)^3(x-4)^4$. This defines a function $f : \mathbb{R} \to \mathbb{R}$. There is a unique natural number $k$ such that every antiderivative of ...
0
votes
1answer
69 views

Verify the polynomial of degree $≤ 4$ is exact: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$. [closed]

Verify that the following formula is exact for polynomial of degree $≤ 4$: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$. I'm not sure how to go ...
1
vote
1answer
31 views

Indefinite integral: chance of analytic solution?

Is there a chance the following indefinte integral has an analytic solution? $I(a, b) = \int \mathrm{d}x \sqrt{a^x +b^x}$ My attempts and those of Mathematica proved fruitless thus far... Any ...
0
votes
0answers
15 views

How to integrate a distribution in spherical coordinates over a circle?

I have an angular distribution $\frac{s \sigma}{d\Omega} = \frac{d\sigma}{d(cos\theta)d\phi}$. How can I calculate it over a circle which lies on the plane $X = dist$, has radius $r$ and its centre is ...
0
votes
2answers
75 views

Find $f$ if $ f'(x) = \frac{2x}{\sqrt{1-4x^2}} $ [closed]

How can we find $f$ if its derivative is: $$ f'(x) = \frac{2x}{\sqrt{1-4x^2}} $$
0
votes
0answers
35 views

Vector integration problem

Any help will be appreciated for the following question: Question: Represent $\displaystyle \int_S \nabla \times \overrightarrow{F} \cdot d\overrightarrow{s}$, where $S$ is the hemisphere ...
1
vote
0answers
44 views

What is the Beta-Function used for in Calculus?

For a Calculus I course we have to learn about the Beta-function which is defined by $$B(q,p) = \int_0^1x^{p-1}(x-1)^{q-1}dx$$ What is the use/application for this? We are only given it as-is without ...
1
vote
2answers
40 views

Derivatives and Integration [closed]

What is my rule? I have a y-intercept of 3 and at x=-2 my value is -45 and my slope is 61. My derivative crosses me on the y-axis. If the derivative of my derivative is a linear function i.e has the ...
0
votes
1answer
19 views

Need to know if the given du value for this integral/ln solution is correct

$\int \frac{2}{x(\ln x+5)^7} dx$ for that problem the example gives $u=\ln x+5$ and $du=\frac{1}{x}dx$. based on a couple other problems I thought du should be $\frac{1}{x+5}dx$ The derivative of ...
0
votes
1answer
19 views

Normalizing a biexponential function so its time integral is always unity

I'm reading a paper that uses the following function: \begin{align} f(t) &= \frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2}). \end{align} The authors state that The normalization adopted ...
2
votes
1answer
41 views

Does Pointwise convergence imply $\int_a^b (f_n(t) - f(t))^2 \, \mathrm d t \to 0$

If $f_n$ and $f$ are integrable and $f_n$ tends to $f$ uniformly on $[a,b]$ then $\displaystyle \int_a^b (f_n(t) - f(t))\,\mathrm d t \to 0 $, since $$\int_a^b |f_n(t) - f(t)| \, \mathrm d t \le ...
1
vote
1answer
19 views

Measuring Image of a Set Using Jacobian Integral

Assume $T:\mathbb{R}^d \to \mathbb{R}^d$ is a differentiable mapping and $E$ be a measurable set. Show that $m(T(E))=\int_E |det(DT(x))|dx$. I am thinking I might use the following Theorem, but ...
0
votes
1answer
30 views

How to evaluate double integral over function of square?

I don't understand how to do this question. Can someone show work and explain please? *I attached a picture of the problem because I couldn't figure out how to format it. Picture of problem
0
votes
2answers
53 views

Find $k$ such that $\int_0^{\infty} ky^3 e^{\frac{-y}{2}}dy = 1.$

I'm trying to solve for $k$ given that the integral $$\int_0^{\infty} ky^3 e^{\frac{-y}{2}}dy = 1.$$ I can see that I can pull out k to get $$k \int_0^{\infty} y^3 e^{\frac{-y}{2}}dy = 1.$$ However, ...
27
votes
7answers
3k views

Why does the fundamental theorem of calculus work?

I've known for some time that one of the fundamental theorems of calculus states: $$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$ Despite using this formula, I've yet to see a proof or even a ...
3
votes
1answer
103 views

Complex integration over branch cut

$$ \int_{0}^{\infty}\frac{1}{1+x^{3}}{\mathrm{d} x} $$ I've been able to solve this integral by the residue theorem using a contour along a 1/3 circle and a line along the x-axis, but apparently it ...
1
vote
2answers
51 views

How do I do this integral?

How do I integrate $$\int\limits_{-c}^c \frac{e^{-\gamma\, x^2}}{1-\beta \, x}\, dx $$, under the constraints (if necessary) that $\beta < 1\, /\, c$ and $\beta \ll 1$ ?
2
votes
1answer
45 views

Proof of Reduction Formula for $\int_{0}^{\infty} x^n e^{-x^2} dx$

I have attempted both substitution and integration by parts upon the following problem (and both in conjunction) to no avail. The reduction formula given remains quite elusive. Let ...
0
votes
1answer
18 views

Limits of iterated integrals?

The region for my iterated integral is restricted by $y = 0$, $x=1$, and $y=x$. My book says that if we look at the region as horizontally simple and make horizontal rectangles, we have the integral ...