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 ...

learn more… | top users | synonyms (3)

0
votes
1answer
45 views

Gauss-Legendre three point rule

Use the change of variables $$x=\frac{a+b}{2}+\frac{b-a}{2}t,$$ to show that $$\int^b_a f(x) \ dx = \frac{b-a}{2} \int^1_{-1} f\left( \frac{a+b}{2} + \frac{b-a}{2}t \right) \ dt.\tag{1}$$Hence ...
1
vote
1answer
150 views

question on separable differential equations

I'm trying to solve this equation but always got stuck in a term. How can I separate the $x$,$y$ variables in the equation? $$(y^2 -yx^2)dy + (y^2 + xy^2)dx =0$$
0
votes
0answers
28 views

Finding the area of a circle that is formed by cutting a sphere.

Say I have a sphere $x^2+y^2+z^2=a^2$ and a plane $x+y+z=b.$ How do I find the surface area of this surface? I think I would use the surface integral and for graphs the surface "element" is ...
1
vote
0answers
24 views

finding the Lebesgue measure of the set $A = \{ (x,y,z) \in R^3 | y^2 + z^2 \le 4, 0 \le x \le 4, x \le 6-y-z \}$.

I want to find the Lebesgue measure, denoted $u$, of this (closed and thus measurable) set. $$A = \{ (x,y,z) \in R^3 | y^2 + z^2 \le 4, 0 \le x \le 4, x \le 6-y-z \}$$ This is a cylinder cut by a ...
0
votes
1answer
37 views

Proving integration formula involving the form a+bx

While trying to memorize and understand various integration formulas, I came across an integration rule stating that $$ \int \frac{1}{x^2(a+bx)^2} dx = ...
5
votes
1answer
60 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
2
votes
3answers
32 views

Integral conversion to polar coordinates - bounds

I have an integral $$\int_0^1 \int_0^1\sqrt{x^2+y^2}\ dxdy $$ and its result is $\approx0.765...$ I convert it to polar coordinates and get $$\int_a^b \int_c^dr\ drd\phi $$ But how can i compute ...
2
votes
4answers
105 views

$\int_{-a}^{0}f(x) dx=\int_{0}^{a}f(x) dx$ for an even $f$

$f:\mathbb R \to \mathbb R$ integrable (but not necessarily continuous) on $\mathbb R$ function such that $f(x)=f(-x) , \forall x \in \mathbb R$. I need to show that $\int_{-a}^{0}f(x) ...
2
votes
4answers
103 views

Finding the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. Cylindrical coordinates.

I would like the answer to preferably be done using either using a surface integral, or an integral with substitutions. But anything other than this is alright, if nothing else exists. I have to find ...
2
votes
4answers
86 views

Prove if $f:[a,b]\rightarrow \mathbb{R}$ is continous and not everywhere zero, then $\int_{a}^{b}f^2(x)dx>0.$

Claim: If $f:[a,b]\rightarrow \mathbb{R}$ is continuous and not everywhere zero, then $\int_{a}^{b}f^2(x)dx>0.$ I have the following theorem that seems applicable here - more specifically the ...
2
votes
0answers
33 views

Spherical, polar coordinates, volume of set.

Find the volume: $$\{(x,y,z)\mid x^2+y^2 \leq (z-1)^2 \leq 4-\frac{x^2}{2} - 2y^2, z\geq 1 \}$$ I've got the intersection of the following two basically: \begin{align} 1. & & & (z-1)^2 ...
1
vote
1answer
46 views

Error for Trapezoidal Rule in multi-variable integrals

For one dimension integrals $\int_{a}^{b}f(x)dx $, we know the global truncation error goes like$\ \approx\mathcal{O}(h^2)$ where $h=\frac{b-a}{N}$ and N is the number of intervals. Also knowing how ...
3
votes
3answers
87 views

Differentiation under the integral sign for $\int_{0}^{1}\frac{\arctan x}{x\sqrt{1-x^2}}\,dx$

Hello I have a problem that is: $$\int_0^1\frac{\arctan(x)}{x\sqrt{1-x^2}}dx$$ I try use the following integral $$ \int_0^1\frac{dy}{1+x^2y^2}= \frac{\arctan(x)}{x}$$ My question: if I can do ...
0
votes
0answers
38 views

How to evaluate I(y) = $\int_0^t e^{ax^b} e^{-cx^d} x^f dx$ in terms of special functions?

To put the above in the proper context, I am trying to solve a Bernoulli equation of the second order: $\frac{dy}{dt} = -\frac{A}{p-q}(e^{-pt}-e^{-qt})y-Be^{-rt}y^2$ where constants A, B, p, q, r ...
1
vote
2answers
31 views

Convergence of integral power of $\cos$

Find the integral and find $k$ in order to converge ($k$ is real number). $$\int_0^{\frac{\pi}{2}} \cos (\theta) ^{2k} d\theta.$$ I can find the value of integral if $k$ is integer, but what happens ...
0
votes
1answer
36 views

Solving a differential equation with only one variable

So for some reason I'm blanking on solving a seemingly easy question: $(8x + \cos(x))\mathrm dx + (4x^2 + 7\sin(x) + 1)\mathrm dy = 0$ Which I'm currently assuming is going to be solved using an ...
0
votes
1answer
461 views

Why i got negative value for volume?

I want to find the indicated volumes under the surface $z=\frac{1}{y+2}$ and over the area bonded by $y=x$ and $y^2+x=2$. After sketching the graph for $x=2-y^2$, and $x=y$ i found that $y=0$ and ...
3
votes
0answers
24 views

Some sort of generalized Jensen inequality?

Let $(X, \mathcal{A},\mu)$ a measure space such that $\mu(X) > 0$ and let $f, g : X \rightarrow (0,\infty)$ be such that $f, g, f\log(f), f\log(g) \in L^1(\mu)$. Show that $$ \|f\|_1\log ...
2
votes
1answer
80 views

Help evaluate $\int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx$.

I am trying to evaluate $$ I = \int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx $$ where $a \ge 0$ and $b> 0$. $$ I = \frac{2}{\sqrt{\pi}}\int_0^\infty \int_{a + b\ln (x)}^{\infty} ...
1
vote
4answers
122 views

Error in solving $\int \sqrt{1 + e^x} dx$ .

I want to solve this integral for $1 + e^x \ge 0$ $$\int \sqrt{1 + e^x} dx$$ I start by parts $$\int \sqrt{1 + e^x} dx = x\sqrt{1 + e^x} - \int x \frac{e^x}{2\sqrt{1 + e^x}} dx $$ Substitute ...
0
votes
2answers
86 views

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$?

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$? My book used differentiation under the integral sign to evaluate an integral. The integral was ...
7
votes
1answer
215 views

Integrate $\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
0
votes
1answer
34 views

Is the Riemann integral defined with partitions with subintervals of the same legth different from the general case?

When considering Riemann sums we partition the closed interval $[a,b]$ in subintervals that don't necessarily have to have the same length. Then the Riemann integral is defined by taking the supremum ...
2
votes
0answers
32 views

Solving a 1D integral with system of equations for retarded electromagnetic fields

I need to solve the following integral to calculate the effect of retarded electromagnetic fields on a test charge: ...
0
votes
1answer
36 views

Show that $\int_{\tau}^{\infty}\frac{\exp(u)}{[1+\exp(u)]^2}du=\frac{\exp(-\tau)}{1+\exp(-\tau)}$

How $$\int_{\tau}^{\infty}\frac{\exp(u)}{[1+\exp(u)]^2}du=\frac{\exp(-\tau)}{1+\exp(-\tau)}$$? My Attempt : let $z=1+\exp(u)$ $\frac{dz}{du}=\exp(u)$ $$ \begin{array}{c|c|c|} ...
0
votes
2answers
16 views

About the interpretation of line integrals

I've been asked to compute the line integral of the function $f(x,y)=xy$ over the elipse $\frac{x^2}{4}+y^2=1$ counterclockwise orientated. My doubt is if this means that i have to compute the surface ...
2
votes
1answer
53 views

Integrating a Complex Exponential Function

Suppose $w=\exp(2i\pi/3)$. How would I go about integrating $$\int\frac{3dx}{e^x+e^{wx}+e^{w^2x}}$$ Is there a transformation i can use? This is an entire function; there is no $x$ that will ...
3
votes
2answers
207 views

Lebesgue Integral, existence, improper integrals, etc.

Problem: At the request of another user, I am taking an older question and specifically addressing one problem. I am self-learning about Lebesgue integration, and am just starting to try and apply ...
0
votes
1answer
19 views

Delta function of two variables in integral of three variables

I want to integrate a function like: $$ Z = \int\int\int dt dt' dt'' \, F(t'',t)G(t,t') \delta(t-t')$$ where $F$ and $G$ are arbitrary functions. Clearly we must have $G(t,t)$ in the expression due ...
1
vote
1answer
32 views

Checking whther the integral $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ convergent

I need to check convergence of $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ . I think it divergence cause it bigger than $\int_1^∞ \frac{1}{x} dx$ but I can't prove it. I have an hint that ...
1
vote
1answer
874 views

For each of the following integrals find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral.

$$ \int x\sqrt{7x^2+42x+59} dx $$ There were never any examples quite like this in class, so I'm clueless as to how to figure out which trig function to use.
1
vote
2answers
887 views

Integral of Derivative squared

For a function $f$ I know that: $$\int{f'(r)dr}=f(r)$$ where $f(r)$ is known. knowing the result of this integral how can i calculate $$\int{(f'(r))^2dr}$$ Is there any relation between these ...
0
votes
1answer
29 views

Integral problem, evaluating the substitution at zero…

So I have to show the following: $$\int_0^v \frac{x^a}{(x+k)^{2a+2}} dx = \int^{\infty}_{k^2/v} \frac{u^a}{(u+k)^{2a+2}}du$$ By making a suitable substitution. Where $k>0$ and $a$ is a positive ...
13
votes
4answers
6k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that I have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
1
vote
1answer
56 views

$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$

How can I prove $$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$$ given a measure space $(\Omega,\mathfrak A, \mu)$, a non-decreasing sequence $(f_n)$ of measurable functions on ...
2
votes
1answer
36 views

What is running integral? [closed]

My question might be too simple. But I could not find any source giving the answer. Can you please explain the running integral?
0
votes
2answers
29 views

Indefinite integration of a fraction with a non-factorable denominator

Solve the integral below: $$ \int \frac{x+1}{x^2-4x+6} \, dx $$ I tried u-sub and got $$ u=x^2-4x+6 $$ $$ du = 2x - 4 dx \leftrightarrow dx = \frac{1}{2(x-2)}du $$ $$ \int \frac{x+1}{u} ...
1
vote
0answers
29 views

Help with simple integral

Id like to know why this is wrong: $\int sen(x)·cos(x) dx$$\underset{\uparrow}{=}\int u\ du=\frac{u^2}{2}+c=\frac{1}{2}sen^2(x)+c\\\boxed{CV\\u=sen(x)\\du=cos(x)dx}$ When checking on Wolfram Alpha ...
0
votes
0answers
18 views

Evaluate the integral $f(x,y,z) = x$ within $x^2+4y^2+9z^2 \leq 1$ and $x \geq 0$ and also $y \geq 0$

I am asked to evaluate the integral $f(x,y,z) = x$ within $x^2+4y^2+9z^2 \leq 1$ and $x \geq 0$ and also $y \geq 0$ using a change of variables. Should I proceed with spherical coordinates? If so, is ...
1
vote
2answers
121 views

How to integrate $((x^2-1)(x+1))^{-2/3}$ using the substitution $u=(x-1)/(x+1)$?

I was asked to find the indefinite integral $$\int \frac{1}{((x^2-1)(x+1))^{2/3}} dx$$ using the substitution of $u=(x-1)/(x+1)$. How do I make this substitution? I attempted to solve this ...
0
votes
0answers
183 views

Vector Calculus Temperature Profile

Question : If $T(r) = \frac{T(0)}{r^3}$ is the temperature profile in the region R, then use the previous results to calculate the average temperature in R when $T(0) = 1000$. Verify that the average ...
3
votes
4answers
153 views

Definite integral which does not evaluate

How can I solve the following integral? $$\int_{0}^{2\pi }\frac{\cos\phi \cdot \sin(\theta -\phi )}{1-\cos(\theta -\phi )}\text d\phi $$ I have evaluated this integral on Maple without the limits. ...
0
votes
2answers
27 views

Evaluating the integral of $f(x,y,z) = \frac{y}{\sqrt{z}}$ on $y \geq 0$ and $0 \leq z \leq x^2$ and $(x-2)^2+y^2 \leq 4$

I am asked to evaluate the integral of $f(x,y,z) = \frac{y}{\sqrt{z}}$ on $$ y \geq 0\\ 0 \leq z \leq x^2\\ (x-2)^2+y^2 \leq 4 $$ What I have so far (and it seems a little off) is $$ ...
1
vote
0answers
45 views

Inverse function theorem and integrals.

Consider this integral: $$I = \int g\left(f(x)\right)dx.$$ Assuming all regularity conditions, by inverse function theorem, $$\frac{df(x)}{dx}=\frac{1}{\left[f^{-1}\right]'\left(f(x)\right)}$$ and ...
2
votes
1answer
111 views

Can anyone verify $\int_{0}^{\infty}\frac{e^{-2nx}+2nx-1}{x(e^x+1)}dx=\ln{2n\choose n}$? [closed]

Central binomial coefficient from mathworld $$\frac{2^{2n+1}}{\pi}\int_{0}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx={2n\choose n}$$ Here we have $\ln{2n\choose n}$ in term of another integral, ...
2
votes
2answers
61 views

Sketch the heart and indicate its orientation with arrows $ r = 1 - \cos(\theta)$. Find the area enclosed by the heart

Hi all I am trying to figure out how to sketch the heart. Here is what I have tried so far: $$r = 1 - \cos(\theta) \\ r(r = 1 - \cos(\theta)) \\ r^2 = r - r\cos(\theta) \\ $$ Use the fact that $$r ...
1
vote
3answers
95 views

Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
-1
votes
0answers
43 views

Integral from $0$ to $b$ of the functions $\frac{\cos x}{1+x}$ and $\frac{\sin x}{(1+x)^2}$ [closed]

What is the integral $$\int_0^b\frac{\cos x}{1+x}\,\textrm{d} x$$ And what is the integral $$\int_0^b\dfrac{\sin x}{(1+x)^2}\,\textrm{d} x\ ?$$
0
votes
0answers
33 views

Given $\int f(x)/x\,dx$, seeking $\int f(x)/(A+Bx)\,dx$

Given the value of $\int f(x)/x\,dx$, is there a method to determine the value of $\int f(x)/(A+Bx)\,dx$? In particular, I'm dealing with $f(x)=J_0(Cx)J_1(Dx)$ and integrating from 0 to $\infty$. ...
2
votes
1answer
110 views

Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$ For example $$\int_{- ...