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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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
257 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
621 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
36 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
146 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) \\ &= ...
0
votes
1answer
73 views

Integral of normal distribution curve

I am having hoping to use the integral of the normal distribution curve to find the probability of having a mean of $0.30$ or greater, i.e. one tailed distribution. With a sample standard deviation of ...
3
votes
0answers
63 views

Differentation uder the integral sign

Let $F(x)=\int_{\sin x}^{\cos x} e^{x\sqrt{1-y^2}} \, dy $. My task is to calculate $F'(x)$. My idea is to use http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign and I get: ...
0
votes
1answer
38 views

Boundaries of the triple integral

I need to calculate the triple integral $\iiint_V y\,dx\,dy\,dz$ where $V$ is deterimned by $y\geq x^2, z\leq 4-y, 0\leq z \leq3$. I get $\int_{-1}^1 dx \int_{x^2}^1 y \, dy \int_0^3 dz + 2\int_1^2 dx ...
6
votes
2answers
147 views

Computing a nasty integral (probably with computer algebra system)

I'm trying to do this integral, not sure if it is possible: $$ \int_{1}^{\infty}\int_{0}^{\infty} \exp\left(\, -\,{x^{2} \over y^{2}}\,\right) \exp\left(\,-\,{y^{2} \over z^{2}}\,\right) \exp\left(\, ...
4
votes
3answers
146 views

Dirichlet's integral $\int_{V}\ x^{p}\,y^{q}\,z^{r}\ \left(\, 1 - x - y - z\,\right)^{\,s}\,{\rm d}x\,{\rm d}y\,{\rm d}z$

I found such an exercise: Calculate the Dirichlet's integral: $$ \int_{V}\ x^{p}\,y^{q}\,z^{r}\ \left(\, 1 - x - y - z\,\right)^{\,s}\,{\rm d}x\,{\rm d}y\,{\rm d}z \quad\mbox{where}\quad p, q, r, s ...
0
votes
1answer
70 views

How does an Iterated Integral Work?

I am simply confused because of planes now. Consider: $$J = \int_{R}\int_{R} e^{-(x^2 + y^2)} dxdy$$ What is the geometrical aspect of this integral? This represents the volume under $h(x,y) = ...
0
votes
1answer
69 views

Solving equation with integrals

What is the solution of this integration: $$\int_o^v{vdv\over g-kv/ m}=\int_o^h dy$$ I have tried and my solution goes like this $${m\over k} ln {vdv\over g-kv/ m}=y$$ Any help is appreciated.
0
votes
1answer
38 views

laplace transformation solve heaviside d.e. $y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$

$y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$ I did the transformation and obtained $Y=e^{3s}(\frac{1}{s^2}-\frac{2}{s}-\frac{1}{s^2}+\frac{2}{s+1})+(\frac{3}{(s+1)^2}+\frac{2}{(s+1)})$ This ...
2
votes
2answers
118 views

Asymptotic expansion of double integral

Define $$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} \frac{r\,e^{-r^2/2t}}{\sqrt{1-(\sin\theta\sin r \cos\varphi + \cos \theta \cos r)^2}} \mathrm{d} r \,\mathrm{d} \varphi$$ Clearly, for ...
6
votes
2answers
150 views

Easiest way to calculate $ \int_{0}^{1} \frac{ \log (1+x)}{x} dx$

What is the easiest way to calculate $$ \int_{0}^{1} \frac{ \log (1+x)}{x}\, dx$$ ? Need a hint.
10
votes
4answers
179 views

Substitution for definite integrals

In my experience, Calculus II students dislike changing bounds in definite integrals involving substitution. When facing an integral like $$\int_0^{\sqrt{\pi }} x \sin \left(x^2\right)dx,$$ for ...
0
votes
1answer
35 views

Limits on Sturm-Liouville derivation integrating factor

My course notes have the following exercise: Show that any differential operator $$ A = p_0 (x)\frac{\mathrm{d^2}}{\mathrm{d}x^2} + p_1 (x)\frac{\mathrm{d}}{\mathrm{d}x} + p_2 (x) $$ can be ...
0
votes
1answer
46 views

Function in $\mathcal{L}^p(\mu)$ for all $1\leq p \leq 2$

Let (X, $\mathscr{A}$, $\mu$) be a measure space and f $\in$ $\mathcal{L}^1(\mu)\cap \mathcal{L}^2(\mu)$. Can I ask how to show that $f \in \mathcal{L}^p(\mu)$ for all $1\leq p\leq 2$?
2
votes
1answer
46 views

Integrating factor

Can anyone give me some hints as to how to solve the following question? I have to show that the equation below has an integrating factor of the form $t^2\theta^c$ where $c$ is an integer. ...
1
vote
1answer
47 views

laplace transformation solve heaviside d.e. $y''+4y=U(t-4)$

$y''+4y=U(t-4)$ so that $y(0)=3$ and $y'(0)=-2$ I have applied the transformation in both terms obtaining $Y=\frac{3s^2+10s+1-e^{4s}}{s(s+4)}$. How can i solve it?
2
votes
3answers
103 views

Evaluate $\int\frac{\cos^2x}{1-sinx}dx$

$\int\frac{\cos^2 x}{1-\sin x}dx$ can someone explain me how to solve this one and please show your complete solution? So am I supposed to make the numerator $1+sinx$? but I think that doesn't help. ...
4
votes
2answers
223 views

How prove that $\int_0^\pi {x\,f(\sin x)\,} dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)} \,dx$ [duplicate]

To prove that $\int_0^\pi {x\,f(\sin x)\,} dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)} \,dx$ is true, first I started calculating the integral of the left indefinitely $$ \int {x\,f(\sin x)\,\,dx} $$ ...
4
votes
0answers
76 views

Evaluating $\int_0 ^\infty \frac{\sqrt{x}}{e^x-1}dx$

I was trying to compute: $$ I_{1/2}=\int_0 ^\infty \frac{\sqrt{x}}{e^x-1}dx. $$ I know it can be recast as follows $$ I_{\alpha}=\int_0^\infty \frac{x^\alpha}{e^x-1}\ dx= \int_0^\infty ...
0
votes
0answers
15 views

use laplace transformation to solve $y^{iv}-16y=0$, being $y(0)=1$, $y'(0)=0$, $y''(0)=0$, $y'''(0)=0$

Folowing the process, i came to $Y=\frac{s^3}{s^4-16}$ However, when trying to write the fraction as a sum of other fractions,the system is undetermined. ...
0
votes
0answers
28 views

Implicit assumptions about axes

Suppose you are given, Find the volume of the solid, formed by revolving the graph of: $$x=e^{-y^2}$$ and the $y$-axis $(0 \le y \le 1)$ about the $x$-axis. The problem is, how do you define ...
1
vote
1answer
87 views

integration$\int \frac{e^x}{x}dx$

I want to integrate $\int \frac{e^x}{x}dx$. Attempt 1: Let $u = e^x$, and $dv = \frac{1}{x}dx$ i.e. $v = \log x$. Then by integration by parts, we have $$ \begin{align} \int \frac{e^x}{x}dx &= ...
2
votes
1answer
94 views

How to compute the following integral $\int{\frac{x^2+d^2}{\sqrt{(x^4+b^2x^2+c^2)^3}}\mathrm dx}\,$?

For some time, I have been struggling with the following integral: $$\int{\frac{x^2+d^2}{\sqrt{\left(x^4+b^2x^2+c^2\right)^3}}\mathrm dx}\;,$$ where $b^4-4c^2>0$. I did my best, but I am still ...
1
vote
3answers
65 views

evaluate integral with fractional exponents

Can someone please help me integrating this? $$\displaystyle \int \frac{w\mathrm dw}{(5-3w)^{2/3}}$$ I tried substituting $5-3w = u$ and $-3\mathrm dw = \mathrm du$. So $w = (u-5)/3$ and then we ...
2
votes
0answers
77 views

How do I solve the following integration?

The integral gives probability distribution of $M$, and $M$ is the absolute value of the sum of two random variates both following $P(\mu)$ distribution.
1
vote
1answer
50 views

Volume of solid formed by revolving about $x $= 3

A region in the first quadrant is bounded on the left by the $y$-axis, above by $y=4$ and below by the graph of \begin{equation*} \ y = x^3 \end{equation*} Find the volume of if this region is ...
0
votes
1answer
27 views

Separation of an integral product

Suppose I have the following integral, with $a$, $b$, $\alpha$, and $\beta$ constants and $f(x)$ monotone increasing, continuous, always positive, and bounded on $[\alpha,\beta]$. ...
1
vote
2answers
60 views

Separation of an integral

Suppose I have the following integral, with $a$, $b$, $\alpha$, and $\beta$ constants and $f(x)$ monotone increasing, continuous, always positive, and bounded on $[\alpha,\beta]$. ...
4
votes
1answer
140 views

How can I prove that $\int_{0}^{\infty }\frac{\log(1+x)}{x(1+x)}dx=\sum_{n=1}^{\infty }\frac{1}{n^2}$ [closed]

How can I prove that $$\int_{0}^{\infty }\frac{\log(1+x)}{x(1+x)}dx=\sum_{n=1}^{\infty }\frac{1}{n^2}$$
3
votes
4answers
131 views

Computing $\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$

I want to compute $\int_0^\infty u^{-1}(1-e^{\frac{-u^2 t}{2}})\sin(u(|x|-r))\,du$ and so ,as shown below, I want to compute $$\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$$ Attempt We split ...
0
votes
1answer
70 views

Visualization of relation between integration and derivative operations [duplicate]

If I have a function $f(x)$ and I find the derivative I will get $f'(x)$. Furthermore, if I do the integration of the derivative $f'(x)$, as a result I will get again my original function $f(x)$. ...
3
votes
7answers
106 views

How do I calculate the area $\int_0^2 \frac{1}{\sqrt{2x-x^2}} dx$? [closed]

I have the integral $$\int_0^2 \frac{1}{\sqrt{2x-x^2}}dx$$ I know the answer is $\pi$ but I have problems with limits $0$ and $2$. How can I use $\varepsilon$ in this problem?
0
votes
2answers
80 views

Determine all real polynomial solutions y of a differential equation

Determine all real polynomial solutions y of a differential equation $$y'(x) = 5x^7 + 4x^5 + 3x^3 + x + 8$$ for all real numbers $x$. Any hints for starting this would be greatly appreciated.
0
votes
3answers
144 views

inverse laplace transformation of $\arctan(\frac{4}{s})$

inverse laplace transformation of $\arctan(\frac{4}{s})$ using I was trying use 12 but i couldn't arrive to a solution
4
votes
1answer
105 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
7
votes
4answers
783 views

How do “Dummy Variables” work?

I do not understand how dummy variables work in math. Suppose we have: $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$ How is this equivalent to: $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$ How does this ...
0
votes
0answers
107 views

laplace transformation $\cos^2(3t)$ and $\sin(5t)cos(2t)$

it is asked to transform $\cos^2(3t)$ and $\sin(5t)cos(2t)$ using the results from i think the process might be similar for both of them but i don't know wich result to use. can you help me? ...
1
vote
2answers
77 views

Orientation of multiplying integrals

Consider, $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ The trick is to multiply by $I$ again to get $I^2$ But they often write: $$I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2 - ...
9
votes
4answers
218 views

Ways to prove $ \int_0^1 \frac{\ln^2(1+x)}{x}dx = \frac{\zeta(3)}{4}$?

I am wondering if we can show in a simple way that $$ I=\int_0^1 \frac{\ln^2(1+x)}{x}dx = \int_1^2 \frac{\ln^2(t)}{t-1}dt = \frac{\zeta(3)}{4}. $$ Because the end result is very simple, I suspect ...
0
votes
0answers
63 views

Practical calculation of the derivative of the integral of a function

In this Question, I wanted to make sure that $$\frac{d}{dt} \left(\int_0^{t} \phi(t)dt \right) = \phi(t),$$ and provided $\phi(t)$ is continuous, and that a derivative of the function exists, it was ...
2
votes
3answers
213 views

What is the derivative of the integral of a function?

Is this correct ? $$ \frac{d}{dt} \left( \int_0^t \phi(t)dt \right) = \phi(t) $$ If not, how can I recover $$ \phi(t) $$ knowing only $$ \int_0^t \phi(t)dt $$ ?
2
votes
0answers
69 views

Calculate contour integral (Cauchy integral formula)

I have to calculate (without refering to residue theorem) $$\int_{\partial B(2,3)} \frac{dz}{z^4-16}$$ My attempt: First, I need to find singularities of $f(z)=\frac{1}{z^4-16}$. ...
1
vote
1answer
25 views

How to find the integral of a recursive function

How do I find the integral to a recursive function? For example, given: $a(s) = i + s j + s^2 k$ $s(t_0) = q $ $s(t + \epsilon) = s(t) + \epsilon a(s(t)) $ I want to find $s(t_1)$. I know I want ...
3
votes
0answers
97 views

Integration involving square root function

First , i multiply numerator and denominator by (2-x-x^2)^(1/2), then I split the integral into 2 parts , using trig substitution , part 2 is easy to be integrated .. but when I tried to ...
0
votes
0answers
35 views

An integral $\int_0^\infty \frac{\sin^3x}{x^3}dx$ [duplicate]

$$\int_0^\infty \frac{\sin^3x}{x^3}dx$$ How to calculate this integral in mathematical analysis? I have integrate by parts, but got something also difficult to integrate...