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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0
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1answer
42 views

Are the two integrals equivalent?

Consider $x \in \mathbb{R}$, $A\subseteq \mathbb{R}$, $f(x)$ continuous in $\mathbb{R}$ and the integral $$ g(A):=\int_{x \in A}^{} f(x) dx $$ Is $g(A)$ equal to the integral $$ ...
2
votes
2answers
192 views

What is the easiest way to integrate $y=\frac {x+4}{\sqrt{-x^2-2x+3}}$?

What is the easiest way to integrate $y=\frac{x+4}{\sqrt{-x^2-2x+3}}$ ? I tried to integrate it by making numerator in form: $-2x-2$ and then pulling it under differential, but the result drastically ...
6
votes
4answers
336 views

Does the improper integral exist?

I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...
3
votes
1answer
27 views

Convergence in average on every set implies convergence?

Let's say we're working in a measure space $(X, \mathcal{B}, \mu)$, and let $f_n, f$ be measurable. Suppose I have that, for any measurable set $E$, $$ \int_E f_n d \mu \to \int_E f d \mu $$ Does that ...
2
votes
2answers
204 views

Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
2
votes
1answer
163 views

Integrate $\int^{\ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = ...
1
vote
1answer
65 views

positive measurable function on $[0,1]$

If $f$ is a positive measurable function on $[0,1]$, which is larger, $$\int_{0}^{1}f(x)\,\log f(x)\,dx \qquad \text{or} \qquad \int_{0}^{1}f(s)\,ds\int_{0}^{1}\log f(t)\,dt$$ Can you help me ...
0
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1answer
41 views

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge.

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge. Using Wolfram Alpha: http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605 the result ...
2
votes
2answers
51 views

Integral along $\Gamma_c := \{c + i t \mid c>0 , -\infty < t < \infty\}$

I have a Complex Analysis homework problem which I've been working on for some time, and have become stuck. I am asked to compute $$ I \equiv {1 \over 2\pi{\rm i}}\int _{\Gamma_c}{a^{s} \over ...
0
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1answer
55 views

Help with finding the definite integral of $e^{\frac{2x-x^2}{2}}$?

I have this integral that I am trying to evaluate by hand, but I am encountering some difficulties. According to Wolfram Alpha, the answer seems to be: However, I do not understand how they got ...
2
votes
1answer
139 views

Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$

Hey I am trying to integrate $$ I_n:=\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$ Thanks. This integral is old. I am also looking for literature on these integrals ...
4
votes
3answers
106 views

Limit of the integral of $\frac{x^n+1}{x^n+2}$

Consider the following integral: \begin{align} F(x) = \lim_{n\rightarrow\infty}\int f_n(x)dx = \lim_{n\rightarrow\infty}\int\frac{x^n+1}{x^n+2}dx \end{align} How does one evaluate this integral ...
0
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2answers
74 views

Please, help with this integral.

$$\int \frac{\sqrt{x^2-1}}{x^5\sqrt{9x^2-1}}\; dx $$ Thanks for your help.
1
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1answer
44 views

Prove $\ln\big(\frac{21}{10}\big) \leq \sum_{n=10}^{20} \frac{1}{10n}\leq \ln\big(\frac{20}{9}\big).$

Please help. I have been trying his question for the past 2 hours and cant seem to go anywhere with it. $$ \ln\big(\frac{21}{10}\big) \leq \sum_{n=10}^{20} \frac{1}{n}\leq \ln\big(\frac{20}{9}\big). ...
7
votes
2answers
271 views

Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
1
vote
1answer
93 views

Integral$\int_{-\infty}^\infty x^{2n} e^{-\beta (x^2+\cos x+\alpha x)}dx$

Hi I am trying to integrate $$ \int_{-\infty}^\infty\int_{-\infty}^\infty (xy)^{2n}\exp\left({-\beta(x^2+y^2+\cos x+\alpha x+iy)}\right)dxdy \quad \alpha,\beta,n >0. $$ These integrals can be ...
1
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1answer
46 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
3
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0answers
122 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
0
votes
1answer
124 views

Work done by a vector field along a curve

I have to find the work done by a vector field $\vec{F}$ along a curve $C$ from $t = 0$ to $t = 1$: $$ \vec{F}(x,y,z) = (2xe^y + 2x)\vec{i} + (x^2e^y)\vec{j} + (3z^2)\vec{k} \\ C: \;\vec{r}(t) = ...
3
votes
2answers
98 views

Why this equation is true?

Pardon my ignorance. I don't know enough calculus to understand this. I assume this is a very easy question for this amazing site. I saw this on the The Theory of Riemann Zeta Function Book. ...
0
votes
0answers
46 views

Why is $\frac {dy}{dx}$ treated as a fraction? Plus an implicit differentiation question. [duplicate]

Why is $\frac {dy}{dx}$ treated as a fraction? I always thought that it is just notation for the derivative of $y$ with respect to $x$, but when it comes to implicit differentiation and integration ...
0
votes
3answers
143 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
2
votes
5answers
444 views

Solution of a integral

$$ \int e^x \, \left(1 + \frac{e^{-x}}{x} \right) \,dx $$ I got three different integrals from this one, which are integral of $e^x$, integral of $1/x$ and the third one is integral of $e^{-x}/x$ but ...
0
votes
4answers
37 views

Integrate the given equation

Here is the equation that has to be integrated plus my answer: $$ \int \frac{x}{2x^2-1}dx=\frac12\cdot\frac23\ln{|1-2x|} +C $$ The correct answer: $$\frac14 \ln{|1-2x|} + C$$ How it come? Thanks
0
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1answer
37 views

Integrate the following

Here is the equation that has to be integrated and my ans: $$\int x\left(x^2+1\right)^3dx=\int \left(x^3+x\right)^3=\left(x^3+x\right)^4\dfrac 1{12}+c$$ The correct answer is : $$ \int x(x^2+1)^3 ...
2
votes
2answers
59 views

Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
0
votes
3answers
55 views

Changing limits of integration

Is that true, that for every function $f(x)$ changing the limits for opposite, causes: $$\int^0_{-a} f(x) dx = \int_0^a f(-x) dx$$
2
votes
0answers
56 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
0
votes
1answer
70 views

Example of a Riemann integrable function which is not a simple function

I'm looking for an example of a Riemann integrable function which isn't simple? I know that all simple functions $f: I \rightarrow E $ ( where $I \subset \mathbb{R}$ is an interval and $E$ is a ...
0
votes
0answers
29 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
0
votes
1answer
69 views

Integrate square root of 4th grad polynomials

During some calculations for a program I came upon this Integral which I am not able to solve. I already tried Matlab but it didn't help me. Here is the Integral: $$\int\left(\sqrt{\sum_{0}^{5} 9 ...
1
vote
1answer
42 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
1
vote
2answers
46 views

Learning integration, and have gotten to a certain point. What is the next logical step?

I've learnt the following in terms of integration; What are integrals, and what do they represent? Indefinite integrals as the opposite of derivatives Using the power rule for derivatives to provide ...
0
votes
1answer
44 views

Verify the given function including the integral $e^{-x^2}$

I'm really stuck trying to verify that the given function is a solution of the differential equation. I've attempted applying converting it to polar coordinates but I don't think I'm on the right ...
0
votes
1answer
36 views

Integral $ \int_{-\infty}^{\infty} e^{ \frac{-u^2}{2}} du$ [duplicate]

Could you explain what rules are used to compute this integral? $ \int_{-\infty}^{\infty} e^{ \frac{-u^2}{2}} du = \sqrt{2\pi}$
2
votes
3answers
152 views

Integral of 1/(8+2x^2)

I have been following a rule saying that $$\int{\frac{1}{a^2+u^2}}dx = \frac{1}{a}\tan^{-1}(\frac{u}{a})+c$$ The question is asking for the interval of $$\frac{1}{8+2x^2}$$ Following that rule ...
2
votes
2answers
166 views

Integral of $\frac{-2}{\sqrt{16-x^2}}$

So I am asked to find the anti-derivative of $$\frac{-2}{\sqrt{16-x^2}}$$ First step I took was to make it easier for me to visualise $$\int{-2(16-x^2)^{-\frac{1}{2}}}dx$$ I let $u = 16-x^2$ So ...
0
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0answers
29 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
2
votes
1answer
74 views

Prove $\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$

Prove $$\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$$ Where $a_{n}$ is an increasing sequence of positive terms that goes to infinity. I tried to approach it with $\log(a_{n})$ like a ...
4
votes
1answer
347 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
1
vote
4answers
58 views

Integrating a function with substitution

Totally forgot how to integrate. $$ \int \frac{1}{x^2 \sqrt{x^2+4}}dx$$ Just need a tip, for this what would I use to substitute?
2
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1answer
29 views

Is there a set formula for integration like there is for derivatives?

I know that the derivative of $f(x)$ must be $$f'(x)=\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ We can use this formula to derive the derivatives of some functions like $\sin(x)$. Is there such a ...
1
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0answers
34 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
1
vote
2answers
41 views

Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts?

The question is just as it is on the title: Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts? And in case there is, how can I do it?
0
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1answer
41 views

Primitive of $(e^x+e^{2x})^{-1}$

I am looking for a primitive of $\frac{1}{e^{x}+e^{2x}}$. I there an easy way to find it? Thanks.
1
vote
2answers
120 views

Weird computation error when using fnInt (numerical integral) on TI-84 Plus

Today in Calculus class I was bored so I decided to try and approximate $\pi$ by evaluating $ \left( \displaystyle \int_{-a}^a e^{-x^2} dx \right)^2$ on my calculator for larger and larger values of ...
0
votes
2answers
47 views

Finding the curve length

I am solving some curve length questions and came across this question: $$ x = \frac{y^3}{6} + \frac{1}{2y} ; 2<= y <=3 $$ I know how to solve for the curve length when they provide y equals ...
2
votes
0answers
35 views

Improper integrals in curve length

I am supposed to find the length of curve of the following: $ y = \sqrt{2-x^2}$ ; $0\le x\le 1$ $y =\ln(\cos x) $; $0\le x\le \frac{\pi}{3}$ I followed the directions found from this question : ...
1
vote
1answer
39 views

Line integral segment of parabola

Suppose $$ \vec{F} = \nabla f(x,y) = 6y \sin (xy) \vec{i} + 6x \sin (xy) \vec{j}, $$ and C is the segment of the parabola $y = 5 x^2$ from the point $(2,20)$ to $(6,180)$. Then, what is $$\int_C ...
0
votes
2answers
50 views

Is this integral wrong?

$$ \int(x^2+1)(x+1)dx$$ Per partes, let $$u = x+1; u' = 1 $$ $$v' = x^2+1; v = x^3/3 + x $$ $$=(x+1)(x^3/3+x) - \int1*(x^3/3+x) = (x+1)(x^3/3+x) - \frac{1}{3}\frac{x^4}{4}-\frac{x^2}{2} + C $$ ...