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Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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23 views

Why do we choose the limits for double integrals in this way?

This is a general question which I have been asking (to no avail) for a while around my class. I will ask my question using an example. Evaluate $I=\displaystyle\int_{0}^{1}\mathrm{d}x\displaystyle\...
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1answer
23 views

Vector Calculus integration region.

Evaluate $ \int_0^\sqrt2\int_0^{3y}\int_{x^2+3y^2}^{8-x^2-y^2}dzdxdy$. The method given in the answer booklet was to calculate the integrals one at a time and get a numerical answer and it is quite ...
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0answers
21 views

Normalisation constant for e^(-x^a) for all real values x

I've seen this post here on a very similar problem I have: Normalization constant for $f(x) = \exp(-x^\alpha)$ I am trying to find the normalisation constant for $f(x) \propto e^{-x^a}$ for $-\infty &...
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1answer
59 views

Integration Question from MIT integration competition

How would the integral below be done? $$ \int\frac{e^x(2-x^2)}{(1-x)\sqrt{1-x^2}}dx $$
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3answers
29 views

Initial value problem (solve for velocity)

Gravity is a constant value of -10 metres per second squared. An objects initial velocity is 10 metres per second. Determine an expression for velocity. So... g = 10 m/s^2 g = v'(t) v(0) = 10 ...
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1answer
20 views

How do I solve the volume bounded inside a solid

Find the volume of the wedge in the first octant cut from the cylinder $$ y^2+z^2=4 $$by the $yz$ plane and the plane $y=x$. Indicated in the figure the slice used to compute the volume. I cant ...
2
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2answers
56 views

Evaluate $\int \frac{dx}{\sqrt{\frac{1}{x}-\frac{1}{a}}}$

Evaluate the following integral: $$\displaystyle \int\dfrac{dx}{\sqrt{\dfrac{1}{x}-\dfrac{1}{a}}}$$ Where $a$ is an arbitrary constant. How do I solve this? EDIT: I would appreciate it if you ...
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3answers
53 views

Evaluate $ \int \frac{x^4}{(2-x^2)^{3/2}}dx$

Evaluate the following integral: $ \int \frac{x^4}{(2-x^2)^{3/2}}dx$ I've tried to apply Chebyshev theorem on the integration of binomial differentials. We have $ m=4,a=2,b=-1,n=2,p=-3/2$. $\frac{...
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1answer
17 views

Integrating double integral with spherical coordinates problem with interpret a domain

Hello i have the following problem i am solving integral with spherical coordinates but i am getting wrong answer - i think i am integrating correct so i think the problem is coming from the ...
2
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3answers
50 views

Evaluate $\int \frac{dx}{\left(\frac{1}{x}-\frac{1}{a}\right)}$

Evaluate the following integral: $$\displaystyle \int\dfrac{dx}{\left(\dfrac{1}{x}-\dfrac{1}{a}\right)}$$ Where $a$ is an arbitrary constant. How do I solve this? I tried the substitution $$x=a\...
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3answers
37 views

Need help with a step in the solution of an integral

I found this exercise in this pdf. It is the proof of the 5th exercise $ \int \frac{1}{\sqrt {s^2-1}}ds $ Substitute s^2-1 $ s^2-1 = u^2, 2s ds = 2u du \implies \frac{du}{s} = \frac{ds}{u} $ $ = ...
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0answers
27 views

Indefinite integral form of Hypergeometric Function [on hold]

Is hypergeometric F. Has indefinite integral form?
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0answers
33 views

Change of variable in an integration

I'm working on a project, which requires the following numerical integration: $$\frac{a^j}{\Gamma(j)}\int_{-\infty}^t f(s) e^{-a(t-s)}(t-s)^{j-1}ds$$ for some bounded continuous function $f$. Now, ...
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0answers
36 views

Limits of Integral two possibilities?

I have a quick question regarding the limits of an integral. Let's say that you are integrating a function (it doesn't matter too much which function - at least I think) between $1$ and $9$. However, ...
3
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1answer
20 views

Surface area inside cylinder

Find the surface area of the part $\sigma$: $x^2+y^2+z^2=4$ that lies inside the cylinder $x^2+y^2=2y$ So, the surface is a sphere of $R=2$. It looks there should be double integral to calculate the ...
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0answers
33 views

Why is this equality true? Leibniz Integral Rule

\begin{align} f(u,v) &= \frac{\partial}{\partial v} F(u,v) \\ g(u,v) &= \frac{\partial}{\partial u} F(u,v) \\ \lambda (u,v) &= \frac{\partial}{\partial v} g(u,v) \\ &= \frac{\...
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0answers
39 views

Difficult Laplace Transform Type of Integral

Good afternoon. I have the following integral that I need help integrating; $$F(x)=\int_0^\infty e^{s\left(j-\frac{1}{x}\right)}\left[T_N(s)\right]^{k-j}ds$$ where $T_N(s)$ is the truncated ...
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2answers
48 views

Inequality in double integral [on hold]

Given The square $D=\{(x,y)\mid0\le x\le1 ; 0\le y\le 1\}$, prove that $$\iint_D \left(x^3+y^3\right)^{1/3} \ dA \le \iint_D \left(x^2+y^2\right)^{1/2} \ dA.$$
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0answers
79 views

Argument for a new theory of Integration

Consider the following function $$F(x)=\begin{cases} 2^x & x=A_1\\ x^2 & x=A_2\\ \text{Undefined} & \text{Everywhere Else} \end{cases} $$ Where $A_1$ and $A_2$ are disjoint sets that ...
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1answer
88 views

The infinite series of $x^{n^2}$ [duplicate]

I have some troubles with the following series $$\sum^\infty _{n=0} x^ {n^2}$$ I'm suppose to show that this series is equivalent when $x$ approaches $1$ and $x <1$ to $$\frac{G}{\sqrt{1-x}}$$ ...
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1answer
65 views

Differential vs Derivative

I am trying to teach myself $u$-substitution in preparation for Calculus $2$, and I don't think that I quite understand the difference between the derivative and a differential. This goes on to a ...
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1answer
37 views

Please explain this equation

$f(x,y)$ be continuous in the rectangle $R$ given by $a\leq x \leq b,{\ }\alpha \leq y \leq \beta$. $$v(x,y)=\int_{\alpha}^{y}f(x,\eta)d\eta, \quad u(x,y)=\int_{a}^{x} v(\xi,y) d\xi$$ Applying the ...
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0answers
26 views

Show that $\lim\limits_{R \uparrow \infty}\frac{1}{2\pi}\int_{-R}^{R}e^{-i\mu x}\hat{f}_{ab}(\mu)d\mu = f_{ab}(x)$

Show that$$(1)\lim\limits_{R \uparrow \infty}\frac{1}{2\pi}\int_{-R}^{R}e^{-i\mu x}\hat{f}_{ab}(\mu)d\mu = f_{ab}(x)$$ where $f_{ab}(x)$ and $\hat{f}_{ab}(\mu)$ are: More clarifications: Let the ...
4
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1answer
96 views

Integrals of the form $\int \log(2+2\cos x)^ndx$

$\log$ will be the natural logarithm and $\zeta$ the Riemann zeta function. I'm interested in the following family of integrals: $$ I_n = \int_0^\pi(\log(2+2\cos x))^n\mathrm{d}x $$ Some of the values ...
2
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0answers
45 views

Is my solution of $\lim_{x\to\infty}\frac{\int_0^x (\arctan^2 x )dx}{\sqrt{x^2+1}}$ correct?

Is my solution correct? If not, what is the right way to do this and where is my mistake? $$\lim_{x\to\infty}\frac{\int_0^x (\arctan^2 t )dt}{\sqrt{x^2+1}}$$ (L`Hopital.) $$\lim_{x\to\infty}\frac{F'(x)...
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votes
1answer
34 views

How to calculate a double integral

I’ve got the following integral $$\int\int _D \frac{dxdy}{x+y}$$ D is the region bounded by $x+y = 1, x+y = 4, y=0, x=0$ and I have to use the transformation $x = u-uv, y=uv$ Anyone know what domain ...
2
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1answer
74 views

Is it possible to find a general anti-derivative of this equation?

I have an equation of which I need to find the mean value over time, which is $$\sqrt{a^{2}+2aA\cos(t)+4A^{2}\cos(t)^{2}}$$ In this case, a is 1.42, and A is an unknown constant, and so I wish to ...
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2answers
30 views

How can I change the order of integration in this case?

I was solving this problem and I have no idea how should I "Exchange the order of integration to obtain the desired result." I don't think I can use Fubini's theorem since there is a variable y in the ...
3
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2answers
56 views

Remove even elements of partition of integration set

Suppose I am integrating a continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ in a measurable set $A\subseteq I$, where $I$ is an interval: $$ \int_{A}f(x)dx $$ Now suppose I partition the set $I$ in $N$ ...
2
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1answer
34 views

Using $|r|$ versus using $r$ for double integration

Suppose that I have a double integral where the integrand has variables $x$ and $y$, and I am using the polar substitution $x=r\cos(\theta)$ and $y=r\sin(\theta)$. Suppose the region of integration is ...
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2answers
46 views

Contour integration of a function $(z-a)^{-m}(z-b)^{-n}$

Let $m, n \in \mathbb{N}$. Moreover let's find $a, b$ such that both are inside a contour $C$. I am to prove that $$\int \limits_{C} \frac{1}{(z-a)^m} \frac{1}{(z-b)^n} dz = 0$$ Couchy's theorem ...
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0answers
27 views

References Gauss-Bonnet theorem

I read on Wikipedia about that the Gauss-Bonnet theorem for compact two-dimensional Riemannian manifolds $M$ with boundary $\partial M$ with Gaussian curvature $K$ of $M$ and geodesic curvature $k_g$ ...
4
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2answers
67 views

Integration problem in $\lim_{n\to\infty}\sum_{i=1}^n{{\ln(n+5i)}\over n}-\ln n$

Compute $$\lim_{n\to\infty}\left[\frac{\ln(n+5)+\ln(n+10)+\ln(n+15)+...+\ln(n+5n)}n\right]-\ln n$$ My attempts to this question are showed below $\lim_{n\to\infty}\left[\dfrac{\ln(n+5)+\ln(n+10)+...+...
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0answers
54 views

Tricky integral limit [duplicate]

Please help me to solve the following limit using high-school methods(no approximations or DCT). I was thinking to use the squeeze theorem but I can't find the best boundaries to evaluate it. $$\lim_{...
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2answers
47 views

Miscellaneous Solids. How do I solve this problem? [on hold]

Can anybody help me with this Find the volume in the first octant inside the cylinder $x^2/a^2 +y^2/b^2 =1$ under the plane $z=3x$. Use the given slice in the figure to compute the volume.
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0answers
44 views

Is there a function that integrates to the same constant, regardless of the interval of integration?

I am looking for a function $f:\mathbb R^2\to\mathbb R$ so that for $c\in\mathbb R$ given, $$\int_x^y f(x,s)ds=c$$ for all $x<y$. I suspect that it is not possible, since differentiating with ...
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1answer
40 views

How was this integral with functions as limits solved? [duplicate]

How was this integral solved ? $$\displaystyle \frac d{dy}\int_{g(y)}^{h(y)}f(x,y)dx$$ $$=\int_{g(y)}^{h(y)}\frac \partial{\partial y} f(x,y)dx+f(h(y),y)\frac{dh(y)}{dy}-f(g(y),y)\frac{dg(y)}{dy}$$ ...
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1answer
68 views

Evaluation of integral $\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$ / specific characteristic Function

I want to calculate the value of $$I(x) :=\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$$ where $x\in \Bbb R$. Of course we can write $$I(x) = \int_{-\infty}^{\...
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1answer
20 views

If the set of nonequalities of two integrable functions has zero measure they have the same integral

$f,g:[a,b]\to\mathbb{R}$ are integrable and $X=\{x\in [a,b]: f(x)\neq g(x)\}$ has measure zero. Show that $\int_a^bf(x)dx=\int_a^bg(x)dx.$ I proved that each set of measure zero has empty ...
3
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1answer
48 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
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3answers
65 views

Complex analysis integration with contour

Show that the next limit exists and finds its value $$\lim_{R \to\infty}\int_{-R}^{R} \frac{\sin(x)}{x-3i}\ dx$$ My idea is to multiply by the conjugate in the denominator to obtain: $$\int_{-R}^{R} \...
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2answers
38 views

What does the lower integral mean?

Defintion: ${}_{-}\int_a^b f(x) \,\text{d}x = \sup \{L(f,p) \mid \text{$p$ is a partition of $[a,b]$} \}$ where $L(f,p)$ is the lower sum. My problem: I am confused with the definition itself of the ...
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1answer
45 views

Finding the function using fundamental theorems of calculus [on hold]

If the integral $$ \int_{-2x}^3 f(t) dt = \frac{1}{x^2 + 1} $$ find $f(\frac{1}{4})$
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2answers
49 views

Integration by Substitution, Converting dx to du

Q: Using the substitution of $ u = \dfrac{e^{2x}}{5}$, write this integrand as a function of u: $$\int \frac{e^{2x}}{\sqrt{25-e^{4x}}}dx$$ I'm stuck at substituting u into the denominator. The best I ...
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0answers
89 views

More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$

I. In this post, the OP asks about the particular log sine integral, $$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
0
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1answer
24 views

Definite integral involving modified bessel function of first kind, exponential of higher order [on hold]

Is there a solution of integral of the following form: $$\int_0^\infty e^{-x^n} J_0(x) dx $$ I have tried searching a lot but cannot find anything. Any kind of help will be appreciated.
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4answers
54 views

Finding the integral of $x\sec(2x)$ using integration by parts

I need to find the integral of $\int x\sec(2x)dx $, but if I use integration by parts, I get an answer of $\int x\sec(2x)dx =0 $, which I know isn't correct. First I let $u=\sec(2x)$ and $v'=x$. ...
0
votes
1answer
40 views

How to use half angle trigonometric substitution on x/cos(x)

I have this integral that I want to solve for homework: $\int \frac{x}{\cos\left(x\right)}\mathrm{d}x$. After some research, I found out that I can use a half-angle substitution to solve similar ...
1
vote
3answers
83 views

Strange Integral $\int \frac{1}{x\sqrt{2x^{2}-1}}dx$

I've tried countless different methods but I can't get this to work... $\int \frac{1}{x\sqrt{2x^{2}-1}}dx$ How would one go about solving this integral?
0
votes
1answer
30 views

Find $\iiint_V z$ with $V=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$

Let $f(x,y,z)=z$ and $T=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$ Find $\iiint_T f(x,y,z) dV$ I'm having a few problems with this integral, here's ...