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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2
votes
2answers
68 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
vote
3answers
85 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
vote
3answers
119 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
vote
2answers
54 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
74 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
vote
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
vote
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
vote
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
vote
5answers
86 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
50 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
21 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
votes
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
vote
0answers
27 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
59 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
58 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.
-1
votes
0answers
22 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
vote
2answers
53 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 ...
-1
votes
0answers
35 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
vote
1answer
41 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
vote
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
vote
2answers
39 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
76 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
76 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
26 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
47 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
96 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} ...
1
vote
2answers
73 views

What will be $F'(x)?$

Let $F:(0,\infty)\to \mathbb R$ be defi ned by: $F(x)=\int_{-x}^{x} ((1-e^{-xy})/y) dy$ What will be $F'(x)?$
2
votes
0answers
24 views

log-sobolev inequalities for infinite measures.

I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. I tried playing ...
2
votes
2answers
54 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$? According to Serge Lang, the integral on the left is the error term for Stirling's ...
0
votes
0answers
24 views

Theoretical justification for separable differential equations using substitution.

Usually when I have solved separable differential equations, I have just followed a recipie and never understood why I am allowed to do what I do. I see that the heart in solving these equations is ...
1
vote
0answers
33 views

Shell Method About Y-Axis

In my calculus course, we just covered the Shell Method and its uses. I have been doing the homework for a few hours and I am absolutely stumped by a question. The question states: Find the ...
1
vote
2answers
26 views

Application of Integral: Work [closed]

A 360-lb gorilla climbs a tree to a height of 20ft. Find the work done if the gorilla reaches that height in 10 seconds.
2
votes
1answer
79 views

Integral $\int^{1}_{-1} \frac{ln(ax^2+2bx+a)}{x^2+1}dx$ if $a>b>0$

I am trying to evaluate the following integral: $$\int^{1}_{-1} \frac{\ln(ax^2+2bx+a)}{x^2+1}dx,$$ where $a>b>0$. I can't really think of a way to find it so please give me a hint.
1
vote
1answer
31 views

Sum of continuous $L^{1}$ function over the integers.

Let $f$ be a continuous $L^1$ function defined on $\mathbb{R}$, such that $\hat{f}(k) = 0$ for $|k| > 1/2$, where $\hat{f}$ is the Fourier transform. Is it true that $\sum_{k = ...
0
votes
1answer
27 views

If $y=\int_0^x f(t)\sin[k(x-t)]dt$, then calculate $\frac{d^2y}{dx^2}+k^2y$

If $y=\displaystyle \int_0^x f(t)\sin[k(x-t)]dt$, then calculate $\dfrac{d^2y}{dx^2}+k^2y$ $y=\displaystyle \int_0^x f(t)\sin[k(x-t)]dt$ $\implies$ $\dfrac{dy}{dx}=\displaystyle \int_0^x ...
0
votes
3answers
47 views

Compute $\int\frac{1-x}{(x-2)(x+3)}$ and $\int\frac{cos(3x)}{sin(3x)}$

Compute $\int\frac{1-x}{(x-2)(x+3)}$ and $\int\frac{cos(3x)}{sin(3x)}$. I have no idea how to solve these 2 integrals, I've run out of ideas. The first one especially, I can't even start, not sure how ...
0
votes
2answers
45 views

Compute $\int_{1}^{e}\frac{1+\log x}{2x}dx$

I've been trying to solve this integral: $$\int_{1}^{e}\frac{1+\log x}{2x}dx$$ I used a new variable to solve this; $1+\log x = t$ therefore $dx = x dt$, then I inserted this into the original ...
0
votes
0answers
26 views

Periodic antiderivative

Let's consider an integral $\int{\ln(a \cdot \sin(x)+1) \cdot dx}=F(x)$. The value of $a$ is choisen so that $F(x)$ is a periodic function. Find $a$. Ideas: If $F(x)$ is periodic with a period $T$, ...
3
votes
0answers
44 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
0
votes
1answer
56 views

Computing the integral of $\int \frac{25x^2}{(x+3)(x-2)^2}\,dx$ [closed]

How would I find the indefinite integral of the expression $$ \int \frac{25x^2}{(x+3)(x-2)^2}\,dx $$ I have tried using impartial differentiation but was unsuccessful.
1
vote
0answers
21 views

Radial function and integral

Let $\Phi:(0,\infty)\rightarrow(0,\infty)$ be an increasing function and $\rho:(0,\infty)\rightarrow(0,\infty)$ is a function satisying the property $$ \frac{1}{C}\leq\frac{\rho(s)}{\rho(r)}\leq C ...