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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2
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1answer
49 views

Change an integral from Polar to Cartesian Form

I'm trying to convert the following integral from Polar to Cartesian form: $$\displaystyle\int_{\pi/2}^{\pi} \int_{0}^{\sin\theta}\,r^2\,dr\, d\theta$$ I think the integral should be: $\int ...
3
votes
1answer
108 views

Finding the Riemann stieltjes integral using partitions

Please take a look at the following example : $α(x)=x$ when $0≤x≤1$ $α(x)=x+2$ when $1< x≤2$ Evaluate $\int_{0}^{2}xdα(x)$ I solved this using the integration by parts formula and got the ...
1
vote
1answer
48 views

Stuck on simple partial integration

$$\int_0^3\frac{|x-y|}9dy=\frac19\left(x-\frac32\right)^2+\frac14$$ Could someone enlighten me regarding this partial integration? I feel like i'm missing something but I dont know what I am doing ...
1
vote
2answers
36 views

Another question about integrable functions with a transform

I am an engineering student, and taking a real analysis course at demand of my advisor, my inexperience in proofs is giving me hard time. I stumbled upon this example, whose proof left as an exercise. ...
0
votes
3answers
87 views

Calculating this integral?

I'm trying to calculate $$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$ as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ...
1
vote
0answers
16 views

Riemann integration-clarification of an answer

I would like to take your attention to the 8th line where it says $S_{1}≤U(Pϵ,f)$. I just can't see how this was arrived at. I would much appreciate if some one could help me out with this because I ...
2
votes
2answers
73 views

Change of variables in $\int_{0}^{+\infty} \frac{x^{-\beta}}{1+x} \, dx$

Let $0 < \beta < \frac{1}{2}$. I cannot figure out which change of variable to use in order to prove that : $$ \int_{0}^{+\infty} \frac{x^{-\beta}}{1+x} \; dx = \int_{0}^{1} ...
4
votes
0answers
70 views

The Leibniz rule in Euler's works

Does anyone know if the Leibniz rule (the method of differentiation under the integral sign), or a variation thereof, has ever appeared in any of Euler's papers? Any references would be appreciated. ...
1
vote
1answer
102 views

Riemann Stieltjes integral when both f and alpha are discontinuous

$ \int_{0}^3 f(x)\,d ([x]+x) $ where $ f (x) = [x] $ if $ 0\le x < 3/2 $ and $ f (x) = e^x$ if $ 3/2\le x \le2 $ I could break this into $ \int_{0}^3 f(x)\,d ([x]) $ and $ \int_{0}^3 f(x)\,d ...
-1
votes
1answer
34 views

Is there a function that is integrable and continuous on $[a,b]$ but that it is not uniformly continuous on $(a,b)$? [closed]

Any hints or exercises that I could use. I do not know how to start this.
7
votes
2answers
116 views

How to show that $g$ attains maximum at $0$ or $1$

Suppose $f:[0,1]\to\mathbb{R}$ is continuous,define $$g:[0,1]\to\mathbb{R},\quad g(x):=\int_0^1|f(t)-x|dt$$ Show that $g$ attains maximum at $0$ or $1$. I don't know how to approach, any hints?
3
votes
2answers
510 views

If $f$ and $g$ are Riemann integrable, are $f\cdot g$ and $f/g$ Riemann integrable?

I do not think they are, but I cannot seem to come up with a definitive answer. I have tried using the "Cauchy criterion" for integrability $$U(f,P)-L(f,P)<\varepsilon$$ Here, $U(f,P)$ is the upper ...
3
votes
1answer
193 views

An interesting integral $I = \int\limits_{-1}^{1} \arctan(e^x)dx $

I solved this interesting integral online: $$I = \int\limits_{-1}^{1} \arctan(e^x)dx $$ Now I tried the substitution $u=e^x$ but it lead me nowhere. I was looking at the following post which was ...
1
vote
2answers
69 views

Does $ \int x^{-2} \, \mathrm{d}{x} $ have a singularity?

How do you integrate $ \dfrac{1}{x^{2}} $ from $ 0 $ to, say, $ a $? Can you get a principal value? What is the divergence: $ + \infty $ or $ - \infty $?
1
vote
3answers
160 views

Evaluate the integral: $\displaystyle \int x \tan^{-1}\ x \,\mathrm{d}x$

Evaluate the integral: $$\int x\tan^{-1}x\,\mathrm{d}x$$ What I have so far: $$u = \tan^{-1}x$$ $$\mathrm{d}u = \frac{1}{1+x^2}\,\mathrm{d}x$$ $$\mathrm{d}v = x\,\mathrm{d}x$$ $$v = \frac{x^2}2$$ ...
1
vote
2answers
58 views

Need help with Integration Using Substitution?

I'm having trouble integrating: $\displaystyle \int\frac{x^{2}+2}{x+1} \ \mathrm{d}x$ I set U equal to $\displaystyle x+1$ and $du=1$. I get to the step $\dfrac{(u-1)^{2}+2}{u}$ . What should I do ...
1
vote
0answers
378 views

Poisson Integral Formula

I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ...
2
votes
2answers
65 views

Evaluate the integral: $\int e^{-θ}\cos3θdθ$

Evaluate the integral: $$\int e^{-θ}\cos3θdθ$$ My attempt: $$I = \int e^{-θ}\cos3θdθ$$ $$u = \cos3θ \implies du = -3\sin3θdθ$$ $$dv = e^{-θ} \implies v = -e^{-θ}$$ $$(*)\int udv = uv - \int vdu $$ ...
4
votes
3answers
155 views

Proving $\int_a^b x^2dx = \frac{b^3 - a^3}{3}$

Can anyone help me prove $\int_a^b x^2dx = \frac{b^3 - a^3}{3}$, the long way? I know exactly what to do, but the algebra involved is just too much for me and I keep making a mistake somewhere and ...
9
votes
2answers
161 views

$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2012}{x}}{\left(1+ \alpha^x\right)\left(\sin^{2012} {x}+\cos^{2012}{x}\right)}\;{dx} $

For $\alpha\in\mathbb{R^+}$, evaluate $$\displaystyle \int_{-\pi/2}^{\pi/2} \frac{\sin^{2012}{x}}{\left(1+ \alpha^x\right)\left(\sin^{2012} {x}+\cos^{2012}{x}\right)}\;{dx} $$ Can I have a hint on ...
0
votes
0answers
29 views

Why primitives allow us to compute integrals?

I learned how to compute integrals. For example, I know that $\int_a^b f(x) \, \mathrm dx = F(b) - F(a)$ where $F(x)$ is a primitive of $f(x)$. But my question is simple and maybe stupid : why ...
3
votes
1answer
88 views

Keyhole integral and version of $\log$ in $\frac{\log t}{(t^2+1)^2}$

I want to calculate $$\int_0^\infty \dfrac{\log t}{(t^2+1)^2}dt$$ It certainly looks like a contour integral. I'm thinking about the keyhole contour where the "hole" is around the origin and along ...
0
votes
2answers
77 views

Derivative of $\int x^5 (x^6 - 6)^4 \rm dx$

Find the derivative of $\int x^5 (x^6 - 6)^4 \rm dx$ I'm not sure how to do this problem with the integral. If you could provide a thorough explanation, that would be great.
1
vote
3answers
114 views

Find the total area between the curve and the x-axis.

Let $$y= \frac{2}{x^2},\;\quad 1 \le x \le 2$$ I'm asked to find the area between the curve and the x-axis. I think we have to use integrals to solve this? I'm not sure.
13
votes
2answers
275 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
1
vote
1answer
63 views

Change order of integral

I've got the following integral: $\int_0^T{\int_{\Omega}{\partial_t u(x,t)dxdt}}$ I would like to change the order of the integral, so that I become $\int_{\Omega}{\int_0^T{\partial_t u(x,t)dtdx}} = ...
1
vote
3answers
70 views

Let $\,f \colon [0,1] \to [0,\infty)\,$ be continuous . Suppose ..

I am stuck on the following problem that says: Let $\,f \colon [0,1] \to [0,\infty)\,$ be continuous . Suppose $$\int_o^xf(t)\mathrm{d}t \ge f(x) \,\,\quad\forall x \in [0,1] \tag{1}$$ Then ...
3
votes
1answer
92 views

Riemann Steiljes Integral when alpha changes

For example if the question was following $$ \int_{0}^2 x\,d \alpha $$ where $ \alpha (x) = x $ if $ 0\le x\le 1 $ and $ \alpha(x) = 3x $ when $ 1< x \le 2 $ Is it correct to solve ...
1
vote
5answers
198 views

Finding $ \int_{0}^{1} 2x\sin(\frac{1}{x}) - \cos(\frac{1}{x})\ dx. $

How do we solve the following integral ? $ \int_{0}^{1} 2x\sin(\frac{1}{x}) - \cos(\frac{1}{x})\ dx. $ I tried to proceed by integration by parts but got stuck.
4
votes
3answers
261 views

How to evaluate this definite integral $\int_0^2(1-x^2)^\frac{1}{3}~dx$

A student asked me to help him calculate this definite integral $$\int_0^2(1-x^2)^\frac{1}{3}~dx$$ Although I have tried almost all the methods I have learned, I can not still do with it. I have tried ...
5
votes
1answer
189 views

Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$

Let $$f(x)=\large \frac{1}{e^x+e^{-x}+2}$$ Compute the Fourier transform of $f$. We can factor the denominator to get $$f(x)=\frac1{(\exp(x/2)+\exp(-x/2))^2}=\frac1{(2\cosh(x/2))^2}$$ I'm thinking ...
2
votes
1answer
64 views

Regarding Cauchy Integral and Cauchy - Goursat Theorem on $g(z)=\int_C \frac{2s^2-s-2}{s-z} dz$

If $C$ is the circle $|z|=3$ $$g(z)=\int_C \frac{2s^2-s-2}{s-z} ds$$ then using Cauchy Integral $$g(2) =\int_C \frac{2s^2-s-2}{s-2} dz = 2\pi i (2(2^2)-2-2) = 8\pi i$$ But what can we say about ...
1
vote
1answer
69 views

$\pi(x)$ asymptotic as integral $1/\log t$

From the prime number theorem we know that $\pi(x)\sim x/\log x$, i.e. $\dfrac{\pi(x)\log x}{x}\rightarrow 1$ as $x\rightarrow \infty$. How can we use that to show that ...
3
votes
2answers
39 views

Limit $(t-1)\zeta(t)$ as $t\rightarrow 1^+$

Show that $\lim_{t\rightarrow 1^+}(t-1)\zeta(t)=1$. For $t>1$, we can use the definition $\zeta(t)=\sum_{n=1}^\infty \dfrac{1}{n^t}$, so it is approximately $\int_1^\infty \dfrac{1}{x^t}dx$. ...
0
votes
1answer
68 views

Help with solving for a flow curve:

So I'm preparing for a final exam in multivariable and our textbook posed the following question: find the flow lines of F(x,y) = (-y, x) Which I can't seem to solve correctly. We are told that a ...
1
vote
2answers
59 views

Fourier series - Integral

Let $f$ be a complex-valued piecewise continuous function defined on the interval $[-\pi,\pi]$ and let \begin{equation} \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos(nx)+b_{n}\sin(nx) ...
2
votes
3answers
86 views

if f'(x)<g'(x) is f(x)<g(x)?

show that : (x+1)ln(x+1)-1$<$$x^2$/2 okay so i want to show that f(x) $<$ g(x) when x$>$0 f(x)=(x+1)ln(x+1)-1 and g(x)= $x^2$/2 (x+1)ln(x+1)-1<$x^2$/2 deriving the functions give ...
2
votes
0answers
82 views

A problem on Riemann Stieltjes Integral

$ \int_{0}^2 x\,d \alpha $ where $ \alpha (x) = x $ if $ 0\le x\le 1 $ and $ \alpha(x)=2+x $ when $ 1<x\le 2 $ I did this by taking a partition which divided the interval $[0,2]$ to $2n$ equal ...
3
votes
2answers
76 views

Try to solve the following differential equation: $2y''=e^y$

I am trying to solve this equation: $$2y''=e^y$$ No $x$ in equation so: $$y''=P'P , y'=p \\ \implies 2P'P=e^y$$ After the integrating on both sides I got: $$P^2=e^y$$ and back to $y$: $$y'2=e^y \\ ...
3
votes
1answer
106 views

Integration by parts with few regularity

I'm having problems proving an integration by parts formula presented in the work of Alt and DiBenedetto on porous media flow (Remark 3.4.2). Essentially, the problem is the following. Let $s\in ...
4
votes
2answers
165 views

Is there such a thing as partial integration?

Recently in my mathematics courses I was taught partial derivatives, and I wondered if the reverse exists for integrals. This may sound like a stupid question, and it probably is, but let me explain: ...
1
vote
1answer
81 views

Bounding an integral $\int_\Omega \frac{1}{|x-y|^{q}}\;dx$

Let $q \geq 0$ be a real number. Let $\Omega \subset \mathbb{R}^n$ be open and bounded. Let $y \in \Omega$. Can someone show me how to bound the integral as follows: $$\int_\Omega ...
1
vote
0answers
15 views

Convergence of integrals in an arbitary Rieszspace

Hello i have a question about integration theory: The situation: Let $X$ be a set, $F$ a Rieszspace of $X$ and $\varphi$ an integral on $F$. Now we can extend this to the space of all integrable ...
1
vote
4answers
84 views

Inverse Trig & Trig Sub

Can someone explain to me how to solve this using inverse trig and trig sub? $$\int\frac{x^3}{\sqrt{1+x^2}}\, dx$$ Thank you.
1
vote
1answer
83 views

How to calculate the integral of a function with a root?

I have to solve this integral: $\int\limits_{-1}^1(3x^3-5x^2+12x-9)~dx$ I used Grapher (a nifty program that comes with Mac OS X) to display the curve of $f(x)=3x^3-5x^2+12x-9$ and it obviously has ...
0
votes
1answer
39 views

Ordinary Differential Equations - Step by step

hi i'm in first year calculus 2 Differential Equations, I'm having trouble with the steps of integrate the trig identity. find the general solution of the equation (1/x)y'= 4 cos (2x)
2
votes
3answers
540 views

Volume of $n$ dimensional ellipsoid

Let $c_1,c_2,...,c_n$ be positive constants. Consider the $n$ dimensional ellipsoid given by $\{(x_1,...,x_n)|\sum_{k=1}^n\frac{x_k^2}{c_k^2}<1\}$. Prove that it's $n$ dimensional volume is ...
2
votes
3answers
133 views

Integral with a limit; integral and inequality

I am trying to solve the following problem. $$ \lim_{h \to 0} \int_0^h\frac{\sqrt{t^2+9}}{h}\mathrm{d}t $$ My presumption is that I should just evaluate the function at $0$, but I can't justify why ...
1
vote
1answer
84 views

Review of calculus course over the break

I am deeply sorry if this thread or discussion topic does not belong to this forum, but I have no idea on where to post this issue of mine. Essentially I have finished a Calculus 1 course, but kind ...
3
votes
1answer
140 views

Parseval's Identity (Integral)

Calculate the integral: \begin{equation} \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx\end{equation} I'm familiar with Parseval's identity which states that for ...