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
votes
0answers
25 views

How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$

want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$. I did this using polar coordinate and got ...
1
vote
2answers
36 views

What does the notation $\int_A$ mean, where $A$ is an event in a probability space?

I am used to seeing integral notation like this, which means the integral over the domain from a to b. $$ \int_{a}^{b} $$ But now I am looking at a statistics book that says "let A be an event" and ...
3
votes
2answers
73 views

Basic integration question.

I have the integral $$\iint x^2y^2 \ dx\,dy$$ but I am meant to evaluate it at the limits $0<y<1$ and $-2y<x<2y$. I am wondering what terminals of integration I should put in for $x$. Do I ...
1
vote
0answers
24 views

How to find the average velocity of blood?

The velocity v of blood that flows in a blood vessel with radius R and length L at a distance r from the central axis is v(r) = P 4ηL (R2 − r2) where P is the pressure difference between the ends of ...
0
votes
2answers
37 views

Exponential Integration [duplicate]

I don't know how to solve this equation: $$\int_0^\infty e^{-x} (x-a)^m dx$$ where $a$ is a constant and $m$ = $n+1$ Thanks in advance for your help.
3
votes
6answers
45 views

How can I prove that $ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c $?

How can I prove that $$ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c? $$ I don’t know how to prove this identity. Any help? I tried to multiply by $ \dfrac{\cosh(x)}{\cosh(x)} $, ...
1
vote
1answer
35 views

The convergence of $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$ with two different tests.

Let $p,q\in\mathbb{R}$ and consider the series $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$. i) Show by the comparison test, that the series is convergent if $p>1$ and divergent if ...
-3
votes
1answer
63 views

What methods to use to integrate $\sqrt{1+t^4}$?

I have this integral to evaluate $$\int^x_1 \sqrt{1+ t^4}\, dt$$ I have tried substitution, trig identity and integration by parts, i don't have any answer. Can anyone explain the method I need to ...
-1
votes
0answers
15 views

Can someone explain how to find the surface area of the unit sphere using Fubini's Theorem? [on hold]

Can someone explain how to find the surface area of the unit sphere using Fubini's Theorem?
2
votes
0answers
15 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
0
votes
1answer
15 views

Solving IVP by Laplace transform

I'm trying to solve an IVP with non-constant coefficients $$ y'' + 3ty' - 6y = 1, \quad y(0) = 0, \; y'(0) = 0 $$ Taking the Laplace yields $$ s^2Y + 3(Y + sY') - 6Y = \frac{1}{s}$$ $$ Y' + ...
0
votes
3answers
29 views

How do I calculate the limit of this integral from n to n+2?

I need to find the limit, as $n\to\infty$ of $\int_n^{n+2}e^{-x^3}dx$. I tried taking the integral using integration by parts but that doesn't work so now I'm stuck.
5
votes
3answers
76 views

How can I finish integrating $\int {\sqrt{x^2-49} \over x} $ using trig substitution?

$$\int {\sqrt{x^2-49} \over x}\,dx $$ $$ x = 7\sec\theta$$ $$ dx = 7\tan\theta \sec\theta \,d\theta$$ $$\int {\sqrt{7^2\sec^2\theta - 7^2} \over 7\sec\theta}\left(7\tan\theta \sec\theta ...
1
vote
3answers
45 views

Simple Integration by Substitution requiring bizarre answer

Before addressing my queries and attemps I shall be posting the full question below. Use the substitution $x=e^u$ to find $$\int (\ln x)^2dx$$ My answer boiled down to $\dfrac{2x^3}{3} + C$ ...
1
vote
1answer
87 views

Find the antiderivative for $f(x)=\frac{1}{1+\cos^2x}$

Evaluate $\int_0^x \frac{dt}{1+\cos^2t}$ $\forall x \in \mathbb{R}$ I got this question in an analysis exam, and I did what everybody does (this), I made $u=\tan t$ and I got ...
0
votes
0answers
44 views

Did I evaluate the improper integral $\int_1^2 {1 \over (1-x)^2}dx$ correctly?

$$\int_1^2 {1 \over (1-x)^2}dx$$ $$u = 1 - x$$ $$-du = dx$$ $$\lim_{a \to 0^-}-\int_0^{-1} {du \over u^2} = \lim_{a \to 0^-}\int_{-1}^{a} {du \over u^2}$$ $$\lim_{a \to 0^-} {1 \over u}|_{-1}^a $$ ...
2
votes
0answers
41 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
0
votes
1answer
18 views

Consider $f(x) =3x^2+2x+a$ where a is parameter such that $\frac{da}{dt}=3$ Let $a =0$, when $t =0$ and…

Problem : Consider $f(x) =3x^2+2x+a$ where a is parameter such that $\frac{da}{dt}=3$ Let $a =0$, when $t =0$ and $A(t) =\int^t_0 \{f(x)\}\,dx$ ( where $\{\cdot\}$ denotes the fractional part ...
1
vote
0answers
18 views

Solving $\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right)$

I got as far as: $$\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right) $$ $$=\frac{2}{\sqrt{\pi}} \int dx \int^{c(x-b)}_0 dy {\sqrt{x^2+a}} e^{-A x^2 - y^2}$$ $$=\frac{-2 c}{\sqrt{\pi}} ...
2
votes
1answer
24 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
2
votes
0answers
18 views

boundary of boundary is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
-1
votes
2answers
35 views

Double integrals- cartesian to polar [on hold]

$$\int^\infty_{-\infty}\int^\infty_{-\infty} \frac{1}{a^2 + x^2 +y^2}\,dy\,dx$$ How can I convert the integral to polar form the hint given in the question is:x-y plane
0
votes
1answer
32 views

Double integral (choice of) change of variables

I'm looking for a way calculate the following integral: $$\iint_D\frac{(x-y)^2(1+2y)}{(1+x+y^2} d(x,y)$$ With $D=\{(x,y)\in \mathbb{R}^2 : 0 \leq x+y^2 \leq 4 \mbox{ and } x\leq y\leq x+2\}$. what ...
0
votes
0answers
27 views

Proving if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$

Prove that if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$. My attempt: Since $f$ is integrable then there's a sequence of partitions ...
-2
votes
1answer
30 views

this is exercise in Rudin chapter 9 Fourier Transform [on hold]

Given $\lambda>0$ and $a\in\mathbb R$, compute $$\lim\limits_{A\to\infty}\int_{-A}^A\frac{\sin\lambda x}{x}e^{iax}dx$$
1
vote
2answers
27 views

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$ where $k(x,y) = \begin{cases} y(x-1), & \text{ $0\leq y<x\leq 1$} \\[2ex] x(y-1), & ...
0
votes
2answers
37 views

Limits and Integration Problem

I have no idea as to how to go about this. Could somebody please help? Let $$\displaystyle ...
0
votes
1answer
17 views

Self similarity function

My self-similarity function is defined by : R(t) = $$ \int_{-\infty}^\infty \mathrm y(x+t)y^\ast(x)\,\mathrm{d}x $$ which is apparently equal to R(t) = $$ \int_{-\infty}^\infty \mathrm ...
0
votes
0answers
16 views

Need help with integrating the function $\frac{-cb}{b(ax+Pbe^{bx})+a}$ w.r.t. $x$

Can someone give me some hints on how to solve the following: $$exp\left(-\int{\frac{cb}{b(ax+Pbe^{bx})+a}\,dx}\right),$$ where $exp(t)=e^{\,t}\,\forall t\in \mathbb{R},e \approx2.71$, and $a,b,c,P$ ...
4
votes
1answer
46 views

$\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$

Question : Solve $\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$ My Try: Let u = $1+\frac{9x}{4}$ Then, $$du = \frac{9x}{4}dx$$ $$dx = \frac{4du}{9}$$ Substituting the above in the main ...
1
vote
2answers
60 views

Evaluating an integral: $\int \frac{1}{(x+\sqrt{x+x^2})^2} dx$

$$\int \frac{1}{(x+\sqrt{x+x^2})^2} dx$$ I don't know how to approach this integral. I tried a few substitutions, but none of them got me to a desirable point.
1
vote
0answers
11 views

Heat Equation and Composition of Functions

Let $u$ be a solution to the heat equation in a domain $U \times [0,T]$. Let $f$ be a $C^2$ function on the closure of $U \times [0,T]$. Assume that $$f = |\nabla f|=0 \text{ on } \partial U \times ...
-2
votes
0answers
26 views

how to solve this equation to put “h” as function of time? [on hold]

I'm trying to get an equation to define the height (h) as function of time. How can I solve this?
0
votes
2answers
21 views

Find monotonic function with definite integral = 1 for a given range

I'm trying to find a function that has some specific characteristics, but I'm a bit stumped. I'm trying to find a continuous function $f(x)$ that uses a given parameter $A$, where $A > 1$. The ...
2
votes
1answer
43 views

Find a rigorous reference that prove the following integration by parts formula in higher dimension?

My professor in the real analysis class had state the following in class but forgot to put the reference of this formula in the power point slide. The formula for integration by parts can be ...
1
vote
1answer
78 views

Integral of $ \int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$

How to solve: $$\int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$$ This question is from a list and it's in the category of problems that involving $\sqrt{x^2\pm a^2}$ and $\sqrt{a^2\pm x^2}$ (triangle ...
-1
votes
1answer
16 views

volume of cylinder using Cartesian, cylindrical and spherical coordinates ? I stopped at the limits of the integration in spherical coordinates ? [on hold]

this is what I did and I think until here I'm ok , i just need the limits of integration in the spherical coordinates . thanks all
-6
votes
1answer
34 views

Find f(x)dx given [on hold]

I don't understand this question I would really appreciate it if someone could please give me the answer and explain how to do it.
-1
votes
1answer
26 views

Why is the mollification $\frac{1}{r^n}\int_\Omega\varrho\left(\frac{|x_0-x|}{r}\right)f(x)\;dx$ of an integrable $f$ infinitely differentiable?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $\overline{B}_r(x_0)\subseteq\Omega$ be the closed ball around $x_0\in\Omega$ with radius $r>0$. Let $\lambda_n$ be the Lebesgue measure ...
3
votes
4answers
113 views

How can I integrate $\int {dx \over x^2\sqrt{x^2 -1}}$?

Let $x=\sec\theta$, so $\mathrm{d}x=\tan\theta \sec\theta\, \mathrm{d}\theta$. Then $$\int {\mathrm{d}x \over x^2\sqrt{x^2 -1}}=\int {{\tan\theta \sec\theta} \over {\sec^2\theta \sqrt{\sec^2\theta ...
2
votes
1answer
39 views

integration by parts exponential

How do you integrate $$\frac{x}{\sigma^2} \exp \left( \frac{-x^2}{2\sigma^2}\right)$$ I have so far tried integration by parts and have gotten stuck. $$u= \frac{x}{\sigma^2}$$ $$du= ...
1
vote
0answers
39 views

How can i evaluate the following limit?

I need to show that the following limit does converge to less than one depending on the values of $f(k)$ and $g(k)$: $$ \lim\limits_{k \rightarrow \infty} \frac{\frac{1}{f(k+1)} ...
3
votes
5answers
115 views

How to calculate the integral $I=\int\limits_0^1\frac{x^n-1}{\ln(x)} \,\mathrm dx$ [duplicate]

How can we calculate this integral: $$I=\int\limits_0^1\frac{x^n-1}{\ln(x)}\,\mathrm dx$$ I believe that integral is equal to $\ln(n+1)$, but I don't lnow how to prove it.
1
vote
0answers
18 views

How to avoid integrating across a singularity in numerical integration?

I'm trying to evaluate expressions of this form: $$f(x)=a(x)\int_0^x\frac{dx'}{a(x')^2}.\tag1$$ Here $a$ is twice differentiable and has some first order zeros, and $f$ is supposed to also appear ...
0
votes
0answers
26 views

Convert Infinite integral to sum

I want to convert an infinite integral to sum. I could not find much info on this online as my integral is from $\infty$ to $-\infty$. For example how would you convert the following? ...
0
votes
0answers
17 views

Definite Integration of a multivariable function [on hold]

$$ \int _{-\infty }^{\infty }\!\int _{ -\infty}^{\infty }\!\int _{ -\infty }^{\infty }\!\int _{ -\infty }^{\infty }\! 4.0\, \left( 0.06003683241\,{\frac {{\it k2}\,\sin \left( 10^{-8}\,{ \it k1} ...
2
votes
7answers
56 views

How can I solve the improper integral $\int_{1}^\infty {dx \over {(x+1)(x+2)}}$

$$\int_{1}^\infty {dx \over {(x+1)(x+2)}}$$ I have the indefinite integral solved for: $$\ln(x+1)-\ln(x+2) + C$$ But I don't know how to finish with $[1, \infty]$.
4
votes
2answers
48 views

How to draw the graph of $f(x)=\int_0^x\left(\frac{t^3-2t^2-4}{t^2+1}\right)\ dt$ using only your calculator?

$$f(x)=\int_0^x\left(\frac{t^3-2t^2-4}{t^2+1}\right)\ dt$$ I need to find the $x$ and $y$ intercepts, and the inflection points of the function $f(x)$ (with both $x$ and $y$ coordinates). I need to ...
4
votes
3answers
55 views

How can I solve the integral $ \int {1 \over {x(x+1)(x-2)}}dx$ using partial fractions?

$$ \int {1 \over {x(x+1)(x-2)}}dx$$ $$ \int {A \over x}+{B \over x+1}+{C \over x-2}dx $$ I then simplified out and got: $$1= x^2(A+B+C) +x(C-2B-A) -2A$$ $$A+B+C=0$$ $$C-2B-A=0$$ $$A=-{1 \over 2}$$ ...
1
vote
3answers
47 views

How do I know when to use partial fractions or long divison with this integral? $ \int {{x^4+1} \over {x(x^2+1)^2}} dx$

$$ \int {{x^4+1} \over {x(x^2+1)^2}} dx$$ Is there a method to determine which way is better?