All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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An integral that might contain a typo

This is one of my homework problems, which I am unable to solve for a while: $$\int x^8\sqrt{7+2x^5} \, dx$$ It seems the integral is not really open to an instant-solve with u-substitution, ...
4
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1answer
66 views

How to integrate $\frac{2+5x^3}{2x^3+2}\sqrt{x^3+1}$

How would one find the integral of $\frac{2+5x^3}{2x^3+2}\sqrt{x^3+1}$ with respect to $x$? I already know the antiderivative (from Wolfram Alpha), but I don't know how to integrate this function ...
4
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1answer
46 views

How to show that $\int\limits_{-\infty}^{+\infty}(n-1)\Phi(x)^{n-2}\phi(x)^2dx$? decreases in $n$?

I was working on a research project that involves taking the integral of $$(n-1)\int\limits_{-\infty}^{+\infty} \Phi\left(x\right)^{n-2}\phi\left(x\right)^2dx,$$ where $\Phi(.)$ is the CDF for ...
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120 views

How do you write an integral and why

A. Year 1 Calculus Student Approach $$ F(x) = \int f(x') dx\, $$ B. Random math paper you find online approach $$ F(x) = \int dx f(x') \, $$ C. Spivak $$ F(x) = \int f(x) \, $$ D. ??? (Edit) ...
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55 views

what are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper regarding active vibration control, where they are called sine and cosine modulated integrals. $y$ is measurement signal with a strong ...
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1answer
93 views

Solving for limit of integration

$$\frac{1}{\sqrt{2\pi}} \int^0_{z_a} e^{\frac{-z^2}{2}} \, dz = 0.48 $$ How would I solve for the value of $z_a$ using a calculator?
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1answer
85 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
4
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1answer
152 views

Find all differentiable functions $f:[0;2] \to \Bbb{R}$ such that $\int_{0}^{2}xf(x)dx=f(0)+f(2)$

Find all differentiable functions $f:[0;2] \to \Bbb{R}$, with $f'$ continuous, such that the function $e^{-x}f(x)$ is decreasing on $[0;1]$ and increasing on $[1;2]$, and ...
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71 views

Definition of integration

The derivative of a function is defined by $$ f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}} $$ provided the limit exists. For example for $f(x)=\sin(x)$ we can prove that ...
4
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2answers
65 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
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1answer
60 views

indefinite integral of inverse trigonometric function

How can we integrate the following I am unable to find a suitable method or formula by which i can get the value of this integral. $$ \int {\sqrt{\cot^{-1} x}} + {\sqrt{\tan^{-1} x}} \, dx$$
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1answer
75 views

Trying to integrate $\int_0^1 x(1-x)(2-x) e^{-(1-x)^2}\ln(1-x)\,dx$

Buenos Dias, Ciao, Hello! My fellow math stack users, I will try to solve this integral $$ \int_0^1 x(1-x)(2-x) e^{-(1-x)^2}\ln(1-x)\,dx $$ I did this $u=1-x$ $$ -\int_0^1 (u-1)u(u+1)e^{-u^2}\ln u \, ...
4
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1answer
60 views

Trigonometric substitution for integral question.

I'm reviewing my quizzes to study for midterm tomorrow, and I came across a problem where I'm supposed to integrate: $$\int\frac{1}{x^2\sqrt{4-x^2}}dx$$ I used Mathematica to solve the problem and ...
4
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1answer
110 views

Integrating $ \int_0^1 \frac{e^{ix}}{x^6+1}dx $

Integrating $$ \int_0^1 \frac{e^{ix}}{x^6+1}dx $$ but having trouble. I factored $x^6+1$ but does not work for the problem. I used identity $e^{ix}=\cos x +i\sin x$, but got nowhere. I can say ...
4
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1answer
76 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
4
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1answer
53 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
4
votes
3answers
166 views

Integrate $(x^2+1)^\frac{1}{3}$

I tried many times to integrate this function using integration by parts, substitution as $x^2+1=t$ without any conclusion... is it integrable? If it is integrable, how can I integrate it? Thank you ...
4
votes
1answer
123 views

Improper integral and existence

If $a,b\in\mathbb{R}, a<b$ and $f:\{x\in\mathbb{R}:a<x\le b\}\to \mathbb{R}$ is a continuous function, define the improper integral $\displaystyle\int^{b}_{+a}f(x)\,dx$ to be ...
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53 views

Trigonometric Substitution

Question: Use the substitution $x=3\sin(t)$ to evaluate the integral of $\int\sqrt{9-x^2}\,\mathrm dx$. I started by making a right triangle and solving for $\sin(t)$ and $\cos(t)$. ...
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1answer
63 views

Convergence of $f(t)= \frac{\exp(it)}{t^a}$

Let $$f(t)= \frac{\exp(it)}{t^a}.$$ For what values ​​of $a$ does the integral $\displaystyle\int_{0}^{+\infty}f(t)dt$ converge? For $a>0$ it's clear with an integration by parts ...
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145 views

Stationary phase method for $\int_{-\infty}^{\infty}f(t)\exp(ix(t^3-t))dt$

I am currenty struggling with the integral $\int_{-\infty}^{\infty}f(t)\exp(ix(t^3-t))dt$ where $f(t)$ is smooth and $f\rightarrow 0$ as $t\rightarrow +-\infty$. I want to obtain the leading ...
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53 views

Volume of revolution over x-axis

Question: I need to find the volume of revolution of $$f(x)=\frac{2}{x+1},\;\; x\in [0,5],\;\;\text{about the x-axis}$$ In order to fully understand this question, one needs knowledge of ...
4
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1answer
62 views

Double integrals in polar coordinates

Determine the domain of $D=\{(x,y) \in \Bbb{R}^2 |x\in [-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}],y\in[|x|,\sqrt{1-x^2}]\}$ in polar coordinates and draw it. Also how would you integrate $$\int\int_D ...
4
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1answer
145 views

Prove $\int_0^x \frac{f(u)(x-u)^n}{n!}du=\int_0^x ( \int_0^{u_n}( \dotsb ( \int_0^{u_1}f(t)\,dt ) du_1 ) \dotsb )du_n$ via IBP

Problem 18-22 on p. 327 of Michael Spivak's Calculus (first edition) is Use induction and integration by parts to show that $$\int_0^x \frac{f(u)(x-u)^n}{n!}du=\int_0^x \left( \int_0^{u_n}\left( ...
4
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1answer
60 views

Evaluating an integral using the Fundamental Theorem of Calculus

I'm taking a Calculus 2 course after not taking Calculus for two years, and I'm having some trouble with some of the basics. I tried integrating this question and got $3/(9/4)^{9/4}$, but when I ...
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1answer
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Surface integral- getting different result using two methods

I'm doing my homework and I came to conclusion I'm not sure is right. I need to find $$\iint_S x dydz+y^2dxdz+z^2dxdy$$ where $S$ is outer side of surface $x=z^2$, and $1\le y \le3$ and $x\le9$. ...
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1answer
138 views

how to show that integral depending on a parameter are continuously differentiable

I'm trying to solve this exercise Let $f:[0,1]\to \mathbb{R} \space$ an integrable function, show: $$g(z):=\int_{0}^1\frac{f(x)}{x-z}dx$$ is a continuous differentiable function on $\mathbb R ...
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2answers
151 views

Show that $\lim_{n\to \infty}\left[ \int_a^bf(x)^n\,dx \right]^{1/n}=\max_{x\in[a,b]}f(x).$ [duplicate]

Suppose $f:[a,b]\to \mathbb{R}$ is continuous and positive. Show that $$\lim_{n\to \infty}\left[ \int_a^bf(x)^n\,dx  \right]^{1/n}=\max_{x\in[a,b]}f(x).$$ My progress: A simpler version of the ...
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Indefinite integral question.

Evaluate the following indefinite integral. $ \int { \frac { 8 }{ 81+{ x }^{ 2 } } } dx $ The answer is $ \frac { 8 }{ 9 } arctan\frac { x }{ 9 } $ I know that it has something to do with this ...
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2answers
73 views

Evaluation of this integral.

What is the result of this integration in a closed form? $ \int{\cos{ x }^2}dx $ We can say $\int { cos\left( 2x \right) dx= } \frac { sin(x) }{ 2 } $ So why we can't say $\int { cos({ x }^{ 2 ...
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1answer
65 views

Passing limit inside integral for functions in $L^1+L^2$ norm

Let $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, and let $f_k$ be functions in the Schwartz class such that $\|f-f_k\|_1+\|f-f_k\|_2\rightarrow 0$ as $k\rightarrow\infty$. Define ...
4
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1answer
56 views

Fundamental Theorem of Calculus Detail

Part One of the Fundamental Theorem of Calculus states that if $g$ is a continuous function on $[a,b]$ that is differentiable on $(a,b)$, and if $g'$ is integrable on $[a,b]$ then ...
4
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1answer
116 views

Understanding the solution of a telescoping sum $\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$

I'm having trouble understanding infinite sequence and series as it relates to calculus, but I think I'm getting there. For the below problem: $$\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$$ The solution ...
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3answers
165 views

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$ The answer given by Wolfram Alpha is $\sqrt{2\pi}\xi\exp(-\xi^2/2)$. Observe how this is related to the Fourier transform of ...
4
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1answer
205 views

Prove Sine integral exists as improper Riemann integral but is not Lebesgue-integrable.

I got to prove that $$\int_0^1 \frac{1}{t}\sin\left(\frac{1}{t}\right)dt,$$ exists as an improper Riemann integral, yet that $$f(t)=\frac{1}{t}\sin\left(\frac{1}{t}\right)\notin ...
4
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1answer
96 views

Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$ \int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1, $$ where $J_2(x)$ is a Bessel ...
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1answer
93 views

Evaluate complex integrals involving cosine

Evaluate the integrals $$\int_{|z|=1}\dfrac{\cos z}{z-3}dz$$ and $$\int_{|z|=10}\dfrac{\cos z}{z-3}dz$$ The first one should be $0$, since the function $\dfrac{\cos z}{z-3}$ is holomorphic in the ...
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1answer
134 views

Evaluating $\int{\frac{1}{\sqrt{x^2+y^2}}\mathrm dx}$

Attempting to calculate $\displaystyle \int{\dfrac{1}{\sqrt{x^2+y^2}}\mathrm dx}$, $$\int{\dfrac{1}{\sqrt{x^2+y^2}}\mathrm dx}=\int{\frac{1}{\sqrt{(y\tan\theta)^2+y^2}}y\sec^2\theta \mathrm ...
4
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2answers
188 views

Solve the following definite integral: $\int_{0}^{\infty}\frac{x^2dx}{({1-x^2})^2}$

Solve the following integral: $$\int_{0}^{∞}\frac{x^2dx}{({1-x^2})^2}$$ I know that substituting some trigonometric functions may help. But I was not able to solve. Can you give me some ...
4
votes
3answers
132 views

Integral of $re^{-r/a}$

How can I integrate $r\,e^{-r/a}$ from $0$ to $\infty$? I integrated by parts, and then was left with something in an indeterminate form. Is there an alternative to solving this without using ...
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4answers
577 views

Integral (square root function multiplied by exponential function) did i do it right?

I'm trying to determine $\int x^3\sqrt{x^2 +1}\, dx$ I said that $u(x) = x^2 + 1$ and then that $dx = 2x\,dx$ so I rewrote the integral as $$\int x^3\sqrt{x^2 +1}\,2x\,dx$$ which is also ...
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1answer
502 views

Integral with Dirac delta (me or wolfram mathematica?)

I tried to compute with Wolfram Mathematica the following integral $$I=\int_0^\pi\int_{-\infty}^\infty x e^{-jx\cos(\theta-\varphi)}f(\alpha\cos(\theta-\psi))\mathrm \, dx\mathrm \, d\theta$$ where ...
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2answers
91 views

Integration of Rational Functions - Problem with proof relating to complex solutions

$\quad$I was reviewing integration of rational functions, and all was going well until I saw this bit, in the end of the explanation: (translated from portuguese by me) $ \qquad \qquad\text{with ...
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1answer
104 views

Matrix-product-integrals?

Whereas the conventional "sum integral" is $$ \lim_{\Delta x\to 0} \sum_i f(x_i)\,\Delta x, $$ a "product integral" is $$ \lim_{\Delta x\to 0} \prod_i f(x_i)^{\Delta x}. $$ Now you're thinking: just ...
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1answer
4k views

Work to pump water from a cylindrical tank

We have a cylindrical tank with radius 2m and length 10m filled with water. How much work does it tank to pump the water out of the tank from the top? My attempt at the problem goes as follow. $g$ ...
4
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1answer
155 views

Difficult Gaussian Integral Involving Two Trig Functions in the Exponent: Any Help?

Here's the integral: $$\int_d^e \exp\left(-a\left((b+c)\cos(x)-\sqrt{b^2 - (b+c)^2 \sin^2(x)}\right)^2 \right) \, dx$$ I've tried using Mathematica: it fails. Can anyone help evaluate it? ...
4
votes
2answers
163 views

Does $\int e^\frac 1x \, \mathrm dx$ has a closed form?

How can i solve the integral $\int e^\frac 1x \, \mathrm dx$? I came across this one while trying to do multiple integral on $\int\int_D e^{(\frac xy)} \, \mathrm dA$ where D is the area between ...
4
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1answer
90 views

Interchanging limit and integration when bounds of integration depend on a parameter

This is yet another question on when it is permissible to interchange limits and integrals. I am interested in the situation when bounds of integration depend on some parameter, and then the limit is ...
4
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1answer
151 views

Integrate the beta function $B(x,y)$ over the unit square.

I was looking a bit around on the site and disovered (to my surprise) that $$ \int_0^1 \log \Gamma(x+t)\,\mathrm{d}x = \log \Bigl(\sqrt{2\pi}\Bigr) - t + t \log t $$ and using this and the fact ...
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2answers
151 views

Finding the area by the graph by the parametric equations.

I know how to find the graphs of an equation using integration. For example: The area of $f(x)=x^2$ from $x=1$ to $x=2$ is $\int^2_1 x^2dx$. Is it possible to find the area of parametric equations, ...