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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1answer
45 views

Finding limit under integral [on hold]

Evaluate if $f \in C [-\pi,\pi]$ $$\lim_{n\to\infty} \int_{-\pi}^{\pi}f(t)\cos(nt)dt$$ and $$\lim_{n\to\infty}\int_{-\pi}^{\pi} f(t)\cos^2(nt)dt$$
2
votes
1answer
78 views

Prove an integral expression equals $\pi\log 2/2$

How do you prove that: $$3\int_0^1 \frac{\tan^{-1}(x)}{x}-2\int_0^{1/2} \frac{\tan^{-1}(x)}{x}-\int_0^{1/3} \frac{\tan^{-1}(x)}{x}-\frac 12 \int_0^{3/4} \frac{\tan^{-1}(x)}{x}=\frac{\pi\log ...
3
votes
1answer
31 views

Computing with Lebesgue integrals

This problem comes from Royden's Real Analysis, 4th ed., pg 84, #19: For a number $\alpha$, define $f(x)=x^\alpha$ for $0<x\le 1$ and $f(0)=0$. Compute $\int_0^1 f$. MY WORK: I know ...
1
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3answers
77 views

Convergence of $\int^\infty_0 \frac{e^{-\sqrt x}}{1+x}dx $

I would like to know if the improper integral $$\int^\infty_0 \frac{e^{-\sqrt x}}{1+x}dx \qquad (1)$$ is convergent or not. I tried substitution and integration by parts but got no simplification. So, ...
0
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0answers
35 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
0
votes
0answers
11 views

Finite Variation Function.

Let $V$ be a right continuous BV function and put $V_t = \int_0^t a_s dC_s$ where $C$ is increasing and right continuous. Is it true that if $V$ is continuous then $\int_0^t |f_s a_s| dC_s < ...
2
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1answer
35 views

Showing a function is in $L^1(\mathbb{R})$

Given that $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$, $1\le p,q\le\infty.$ Define $F(x)=\int_0^xf(t)dt$. How can one show $$(\vert x\vert+1)^{-a}F(x)g(x)\in L^1(\mathbb{R})$$ when ...
1
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2answers
59 views

Why is the derivative changed during trigonometric substitution?

When using trigonometric substitution to find the indefinite integral of an expression, the derivative typically begins as $dx$. Once some expression is substituted for the $x$ in the expression, the ...
2
votes
0answers
31 views

Definite integral of a hypergeometric function of an imaginary argument

How would one deal with such an integral? $$\int_0^\infty\frac{e^{-n r}}{r}{}_1F_1(i/k+1;2;2i kr) \, \mathrm{d} r$$ Here $F$ is the confluent hypergeometric function, $n\in\mathbb{N}$ and $k>0$ ...
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0answers
57 views

How can I solve this integral [on hold]

Let we have the following integral , How can I solve the following integral $$ \int_{}{}\frac{dx}{x^4+1}=?$$
1
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0answers
34 views

How does integration by parts work with multivariable functions

How does integration by parts work with multivariable function? Lets say I have the functions $f(\textbf{x})$ and $g(\textbf{x})$, where $\textbf{x}\in\mathbb{R}^n$. How would integration by parts be ...
0
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1answer
45 views

How to notate higher anti-derivatives?

We can represent the $nth$ derivative of $y$ with the following notation: $$\frac{d^ny}{dx^n}$$ How can we represent the $nth$ anti-derivative of $y$?
2
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1answer
45 views

How to apply the Chain rule when using standard integrals/differentials?

For example, $\frac{d}{dx} \,\arctan x = \frac{1}{1+x^2}$ and I know this because it is given to me in table of standard derivatives and integrals. But if I want to differentiate something like ...
0
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1answer
34 views

Simpson's 3/8 Rule

When deriving Simpson's 1/3 Rule, I used a second order polynomial $P(x) = Ax^2 + Bx + C$, and integrated over the region $[-h,h]$ Integrating gave me: $ \ \dfrac{h}{3}(2Ah^2 +6C)$ I evaluated ...
8
votes
4answers
111 views

Show rigorously that the sum of integrals of $f$ and of its inverse is $bf(b)-af(a)$

Suppose $f$ is a continuous, strictly increasing function defined on a closed interval $[a,b]$ such that $f^{-1}$ is the inverse function of $f$. Prove that, ...
2
votes
2answers
66 views

Is there a better way of writing differentiation and integration?

Differentiation is commonly written simply with a prime mark and an equation, as $(x^2)' = 2x$. Although I find this confusing and think it'd better be written $D(x\mapsto x^2) = x\mapsto 2x$, as ...
1
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3answers
80 views

Find $\int_0^3f(x)+f''(x)dx$

Suppose $f\in C^2[0,3]$ with $f(0)=1$ and $f(3)=3$. Find $\int_0^3f(x)+f''(x)dx$ Integration by parts gives $$\int_0^3f(x)=[xf(x)]_0^3-\int_0^3xf'(x)dx=8-\int_0^3xf'(x)dx$$ I am not sure how to ...
1
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3answers
114 views

Integrating $ \int_0^1 \frac{x-1}{\ln x}\,dx. $ [duplicate]

I need to compute $$ \int_0^1 \frac{x-1}{\ln x}\,dx. $$ Using the fact that $\frac{d(x^y)}{dy} = (\ln x)x^y$, I can't get any clue. Can someone give me a tip on how to better approach solving ...
1
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2answers
51 views

Algebraic Substitution Of Fractions

I already tried to putting the square root like this: $\sqrt{\frac{x}{5 + x}}$ but I dont know what to do next. $$\int \frac{\sqrt{x}}{\sqrt{5+x}}dx$$
7
votes
2answers
73 views

show that $\int_0^{\infty}\sin(u\cosh x)\sin(u\sinh x)\frac{dx}{\sinh x}=\frac{\pi }{2}\sin u$

$$I(a)=\int_0^{\infty}\sin(u\cosh x)\sin(u\sinh x)\frac{dx}{\sinh x}:a>0$$ I started with $$\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-\cos(a+b))$$ so $$I(a)=\frac{1}{2}\int_0^{\infty}\left ( ...
1
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1answer
47 views

Problems with Calculus of Variations lecture material

I'm having trouble understanding the derivation in my Calculus of Variations course material and I was hoping if someone could clarify this out. Here is my reference (as I have rewritten it, the ...
1
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1answer
56 views

How can I prove this integration result?

The question is: how can I prove that: $$\int_{0}^{\pi} \sin^n\theta\ d\theta = \frac{\Gamma\big(\frac{1}{2}\big) \Gamma\big(\frac{1}{2} + \frac{1}{2}n\big)}{\Gamma\big(1 + \frac{1}{2}n\big)}$$
3
votes
2answers
78 views

Determine if the function $1/\lfloor 1/x\rfloor$ is integrable on $[0,2]$

Is this function integrable on $[0,2]$? $$\cfrac{1}{\left\lfloor\cfrac{1}{x}\right\rfloor}$$ I have suspicion that it is, but I'm unsure of how I could determine if that's true.
1
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1answer
30 views

Integral of difference is less than $\varepsilon$

Suppose $f\in \mathscr{R}[a,b]$ which means Riemann-integrable on $[a,b]$, then given $\varepsilon>0$ there is a continuously differentiable function $g$ such that ...
16
votes
4answers
2k views

I can't remember a fallacious proof involving integrals and trigonometric identities.

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: The end result was some variation of 0=1 or 1=2. It involved ...
1
vote
5answers
114 views

Prove that $\int_{1}^{a} \frac 1t dt + \int_{1}^{b} \frac 1t dt = \int_{1}^{ab} \frac 1t dt$

Prove that $$\int_{1}^{a} \frac 1t dt + \int_{1}^{b} \frac 1t dt = \int_{1}^{ab} \frac 1t dt$$ Useful facts: $\int_{1}^{a} \frac 1t dt$ can be written as $\int_{b}^{ab} \frac 1t dt$ Every ...
1
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5answers
83 views

Solving $\int_7^9 \frac{2}{9 + 16x^2}\,dx$ without using trigonometric substitution?

How to evaluate $\int_7^9 \frac{2}{9 + 16x^2}\,dx$ without using trigonometric substitution? I know how to do it with trig substitution, but the problem I'm doing requires me to do it with algebra ...
0
votes
3answers
49 views

Trig Substitution Integral Question

My class is going over trig substitution, but I can't figure this one out, mostly because it's not in the correct form. Could someone help explain how to set up this problem? $$ \int \frac ...
-3
votes
1answer
20 views

Shell Method of $1/x, x=1, x=2$, and $y=0$ about $y$-axis [closed]

Find the volume of the solid bounded by the graphs of these equations. My Attempt: $2\pi \int_.5^1 (?)(1/y)\, dy$
0
votes
1answer
53 views

Convergence of an integral $\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$

For what $\alpha\in\Bbb{R}$ does $\displaystyle\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$ converge ? The $0$ bound doesn't seem to be much of a problem, but I don't see how to deal with the ...
2
votes
1answer
69 views

Why is $\int_\varepsilon^{1/e}-\frac{1}{\\\log(t)}dt = \int_1^a \frac{e^{-x}}{x}dx$

Let $\varepsilon = e^{-a} \in [0,e)$, why does the following hold (including $\varepsilon=0$)? $$ \int_\varepsilon^{1/e}-\frac{1}{\\\log(t)}dt = \int_1^a \frac{e^{-x}}{x}dx $$
0
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0answers
24 views

what does it represent to multiply any $f(x)$ by an interval $[a,b]$?

I know this gives the area under the graph when it is a horizontal function; however, I am trying to prove something related to integrals of any type of a function. Can you tell me if this relates to ...
1
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0answers
26 views

proving that the graph of a function is of Jordan measure zero

Let $f$ be an integrable function from $B$ to $[0,\inf]$ where $B$ is a sphere in $\mathbb{R^n}$. Exercise: For $f$ and $B$, the graph $$ \Gamma=\{(x,f(x)):x\in B\} \subset \mathbb{R}^{n+1} $$ is of ...
0
votes
3answers
58 views

find out the anti-derivative of $(x^2+1)^{-1/2}$

I searched for the anti-derivative of $$ (x^2+1)^{-1/2} $$ and I found that it's $\sinh^{-1}(x)$ or $\ln(x+(x^2+1)^{-1/2})$ and we didn't study yet this function so how can I find the anti-derivative ...
2
votes
1answer
57 views

Collapsing Infinite Integrals

I'm looking into repeated integrals here on Wolfram's Mathworld and I can't seem to figure why it is that the following is true: \begin{align} \underbrace{\int ...
-1
votes
1answer
53 views

If $f$ is continuous and bounded on $(a,b)$, is it true that $f$ is Riemann-integrable on $[a,b]$? [closed]

If $f$ is continuous and bounded on $(a ,b)$ or $(a, b]$ or $[a, b)$, is it true that $f$ is Riemann-integrable on $[a,b]$? Thanks.
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votes
0answers
21 views

Formal language and set theory [closed]

I have to write this formal language example L1(L2L3)=(L1L2)L3 in terms of Set Theory. I'm not sure how to do that. Operations with formal languages can be found here..maybe it's helpfull ...
1
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2answers
52 views

Integral over the unit ball

This question has been asked before, but I did not understand it, so I worked on it on my own and got stuck. Any help would be appreciated. Let $A$ be the region in $\Bbb R^2$ bounded by the curve ...
0
votes
1answer
31 views

what are the properties of the definite integral that are related to inequalities? [closed]

what are the properties of the definite integral that are related to inequalities? I've been searching the internet and asking teachers regarding this seemingly implausible connection, but haven't ...
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votes
0answers
34 views

Integration of $e^{-(3x^2+2\sqrt{2}xy+3y^2)}$ over the plane [duplicate]

How to solve the double integral: $$ \Large \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(3x^2+2\sqrt{2}xy+3y^2)}\,dx\,dy $$ Is it better to transform to polar coordinates?
1
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1answer
40 views

A function such that $f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$ for all $x$

Let $f:\mathbb R\to\mathbb R$ be a function with continuous derivative such that $f(\sqrt{2})=2$ and $$f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$$ for all $x\in\mathbb R$. Find ...
0
votes
1answer
22 views

Using substituion rule for piecewise monotonic function

I am trying to read the ON THE EXISTENCE OF INVARIANT MEASURES FOR PIECEWISE MONOTONIC TRANSFORMATIONS paper from 1973. One has given a map $\tau : [0,1] \to [0,1]$ which is a piecwiese monotonic ...
1
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3answers
80 views

Evaluation of a general trigonometric integral

How can I evaluate the integral $$\int\sin^k(x)\ dx$$ in which I don't know if $k$ is an even or an odd number?
1
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2answers
38 views

Rate of reaction problem involving $\int \frac{1}{k(50-x)(10-x)}\,dx$ [closed]

here is the question I really have no clue how to do it. I think that I need to complete the integration first and sub in numbers provided in the question. Can I get some hints about solving this ...
0
votes
1answer
23 views

Why is the substitution rule different for n>2 and n=1?

In $\mathbb{R}^n$ for $n>1$ the substitution rule for an injective, differentiable function $\phi$ is given by $$\int_{\phi(U)} f(\mathbf{v})\, d \mathbf{v} = \int_U f(\phi(\mathbf{u})) ...
-2
votes
2answers
74 views

Integration of $\frac{x^2+2}{x+2}$

The solutions manual states that A: the integration of $\frac{x^2+2}{x+2}$ is equivalent to B: the integration of $x-2 + \frac{6}{x+2}$ which can then be easily integrated, but can someone explain the ...
1
vote
1answer
73 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
0
votes
1answer
23 views

Compute an integral explicit (or explicit upperbound? Possible?)

I would just like to know if there is an explicit formula/computation for this integral: $$\int_{a}^1 \left(|x-a|^{\alpha} - |x-b|^{\alpha} \right)^2 dx$$ where $\alpha\in (-1/2,0)$ so the integral ...
1
vote
1answer
41 views

Most general setting for the fundamental theorem of curves

I want to learn more about the fundamental theorem of curves. Wikipedia states the theorem for ${\bf R}^3$ only but I found another source (Theorem 5.5.18, in German only) where it is proved for ...
6
votes
0answers
89 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...