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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2answers
70 views

Must $a$ be less than $b$ in the interval $[a, b]$?

If I describe an interval as $[a, b]$. Must $a$ be less than $b$? In my calculus book I read the definition of an integral of a function which is defined and bounded on $[a, b]$. And the area of the ...
0
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1answer
78 views

Complementary Error Function Integral

I'm trying to evaluate this integral: $$\int_{-\infty}^{\infty} \mathrm{Erfc}^2(x) e^{ax+b}dx$$ Initially I am working with $a=1$ and $b=0.$ I've tried integrating by parts in a couple of different ...
0
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1answer
80 views

Need help understanding this integration by substitution: $\int \sec^{2}{x} \tan^{3}{x} \, \mathrm{d}x$

The following is from page 142 in chapter 7 of Barron's E-Z Calculus (formerly Calculus the Easy Way), fifth edition: Evaluate: 43. $y = \int \sec^{2}{x} \tan^{3}{x} \, \mathrm{d}x$ ...
0
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1answer
50 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
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4answers
128 views

Can anyone prove that $\displaystyle \int_0^{\pi}\!0\, dx = 0$ please !? [closed]

Can any body prove that why $\displaystyle \int_0^{\pi} 0\,dx = 0$ !?
0
votes
1answer
25 views

Calculate the area that the following graphs form

I have been trying and trying to solve the following problem (I even used wolframalpha as an extra help, but no success, and I have like 100 calculations in my notebook): The Task: Calculate the ...
0
votes
1answer
75 views

Integration of $\sin(\frac1x)$

I just started to study integrals yesterday so I am not so strong in integrating functions in this moment, however today I met the integral of $\sin(\frac1x)$ and I just can't find its primitive ...
0
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1answer
49 views

Integrating a Matrix in differential-equations

Find $$\int_0^t A(s)ds$$ if $$A(t)=\begin{pmatrix}\sin(t),\cos(t)\\ -\sin(t),\cos(t)\end{pmatrix}$$ I'm a little confused with the format of the question because it asks me to integrate with respect ...
1
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2answers
76 views

Convergence of $\int \log\left(1-\frac{x}{n}\right)^{n}\log(x)$

I'm trying to show that $$\int_0^{\infty}\left(1-\frac{x}{n}\right)^n\log(x)\,dx=\int_0^{\infty}e^{-x}\log(x)\,dx$$ My idea was to apply dominated convergence theorem, since we notice that ...
3
votes
3answers
160 views

How to compute $\int_0^1\frac{t\ln t}{1+t^2}$ ?

How to compute the integral $$\int_0^1\frac{t\ln t}{1+t^2}\ ?$$ So Wolfram alpha says it is exactly $-\dfrac{\pi^2}{48}$ . I tried many substitutions without success, and partial integration as ...
2
votes
1answer
63 views

Integral: Product of sqrt and Gaussian

All, I would have thought mathematica could do this, but it can't. Can anyone think of a good substitution that makes it tractable? Or even do the integral otherwise? $$ \int_{0}^{R}e^{-\beta ...
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votes
2answers
50 views

Integral calculus exp

I have the following integral $\int_0^\infty e^{-6t}~dt=\dfrac{1}{6}$ and I can't remember the properties of integrals or "$e$" to get the result. Can you, please, help me, with the explanation? ...
0
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3answers
111 views

Integrate $3^x$ using the $px + q$ rule

I'm having trouble integrating $3^x$ using the $px + q$ rule. Can some please walk me through this? Thanks
2
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1answer
58 views

how to calculate the area of $S=\{x\in\mathbb{R}|x_1^{2}+x_2^{2}+x_3=4 , 0\leq x_3\}$

How to calculate the area of $$S=\left\lbrace x\in\mathbb{R}:x_1^{2}+x_2^{2}+x_3=4 ,\, 0\leq x_3\right\rbrace$$ I think the difficulty(for me) is to parametrize this function. I thought that it would ...
0
votes
1answer
50 views

Solution of a nonlinear ODE

How can I solve this ODE$$-vU'=2(UU''+(U')^2),$$ where $v$ is a constant. I can see that the right hand side is $(U^2)''$ but is this useful.
1
vote
1answer
125 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
2
votes
2answers
158 views

How to work out an integral with algebra.

I'm a bit of a noob and I am only just starting to learn calculus. I know how to work out a derivative with algebra (instead of using those rules and shortcuts) but I don't know how or if you can do ...
1
vote
3answers
141 views

Find the derivative of this integral: $h(x)=\int_5^{1/x}10\arctan(t)\,dt$

$$h(x)=\int_5^{1/x}10\arctan(t)\,dt$$ Find $h'(x) $. I know how to calculate the derivative of basic integrals,but this one I've been trying to solve for quite a long time,and have not yet ...
3
votes
4answers
145 views

Evaluation of a particular type of integral involving logs and trigonometric function

Is there any closed form for $$ \int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$ if yes then how to prove it?
1
vote
1answer
85 views

Solving a definite integral

How can i find the value of definite integral $$\int_{0}^{\pi}\lfloor\cot x\rfloor dx$$ Here $\lfloor a\rfloor$ means greatest integer value of $a$. My doubt is that $\cot x$ will lie between negative ...
0
votes
1answer
93 views

Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
2
votes
1answer
40 views

Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$.

Question: Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$. Setting the 2 equations equal w.r.t. $z$, $x^2+y^2+1=2-x^2-y^2 \rightarrow x=\pm\sqrt{\frac 12-y^2}$ Therefore the ...
1
vote
1answer
232 views

Riemann integral proof $\int^b_a f(x) \, dx>0$

Prove that if $f$ is a continuous real valued function on the interval $[a,b]$ such that $f(x)\ge 0$ for all $x\in [a,b]$ and $f(x)>0$ for some $x\in[a,b]$ then $\int^b_a f(x) \, dx >0$. The ...
1
vote
1answer
48 views

Compute $\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle\int_0^1\int_1^2\frac{y}{x+y^2}dydx=\int_0^1\int_1^2y(x+y^2)^{-1}dydx$ How do I integrate the ...
1
vote
1answer
169 views

Computing “radius” of the Intersection of a Circle and an Ellipse

I've been stuck on the following problem for awhile now. Does anyone have any ideas as to how to get a solution? Suppose $r > 0$ is a real number. The circle $x^2 + (y + 4)^2 = r^2$ has radius ...
1
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0answers
29 views

Separation of integrand for multivariate integration when integrand is a product of single variable functions

If $f(x)$, $g(y)$, and $h(z)$ are real-valued functions of a single variable, does the following always hold? Is this the case for numerical approximations of the integral using quadrature? $$ ...
1
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1answer
42 views

What is the value of the following integral?

As disscussed in here , we have: now suppose that $X$ and $Y$ are normal random variables with zero-variance limit and zero mean, what is the value of the following integral? ...
3
votes
1answer
122 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...
2
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1answer
81 views

Integrating an exponential times an error function

I have... $$ I = \int_{\sqrt{R^{2}-\left(R-\epsilon\right)^{2}}}^{R}dxe^{-\beta x^{2}}Erf\left[\sqrt{\beta}\left(R-\epsilon-\sqrt{R^{2}-x^{2}}\right)\right]\ $$ (everything's a constant except $x$) ...
10
votes
1answer
144 views

Feynman Parameters

I'm trying to prove the following identity: $$ \left(\prod_{j=1}^n A_j\right)^{-1} = \int_0^1dx_1 \dots \int_0^1dx_n \,\delta\left(\sum_{i=1}^{n}x_i -1\right) ...
2
votes
2answers
87 views

Integrate $\sin(2x)/(1 + \cos^2x)$ with $u$-substitution

Here's how I started: $$\int\frac{\sin{(2x)}}{1+\cos^2 x} dx = -2\int\frac{-\sin x\cos x}{1+\cos^2 x} dx = -2\int \frac{u}{1+u^2}du $$ I know the answer ends up being this from here: $$ ...
2
votes
1answer
30 views

Why holds $\int_{-t}^t\omega_{n+1}(x)\text{ dx}=-\int_{-t}^t\omega_{n+1}(t-x)\text{ dx}$ for the Newton basis polynomials $\omega_{n+1}(x)$

Let $$\omega_{n+1}(x):=\prod_{k=0}^n(x-x_k)$$ denote the Newton basis polynomials and $$x_k:=kh-t\;,\;\;\;h:=2\frac{t}{n}$$ Why holds $$\int_{-t}^t\omega_{n+1}(x)\text{ ...
2
votes
3answers
183 views

a question about double integral

Let $a,b$ be positive real numbers, and let $R$ be the region in $\Bbb R^2$ bounded by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Calculate the integral $$ ...
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votes
0answers
45 views

integral 2D involving complex exponential and cosine

I've some doubts about my solution of this integral: $$I(\phi_{1},\phi_{2})=\int_0^ {2\pi} \,d\phi_{1}\int_0^ {2\pi} \,d\phi_{2} \frac{e^{-in\phi_{1}} e^{-im\phi_{2}}}{2\pi}\frac{e^{il\phi_{1}} ...
3
votes
2answers
84 views

What happens when $\lvert\omega\rvert =1$?

If this is a duplicate in any way, I'm very sorry. I'm brushing up on some Complex Analysis with Special Functions in mind. Here's a problem I'm stuck on. Evaluate the integral $$I=\frac{1}{2\pi ...
0
votes
2answers
65 views

Not sure how to solve this integral

I am taking an online class that has the following homework question. I don't want anyone to solve it for me, but it's been a long time since I've done any integration, and I can't figure out how to ...
4
votes
0answers
89 views

Showing that $\lim_{N \to \infty} \int_{|z|=N+\frac{1}{2}} \frac{ \sinh az}{\sinh \pi z} \mathrm{e}^{ibz} \ dz =0$

To evaluate $ \displaystyle \int_{0}^{\infty} \frac{\sinh ax}{\sinh \pi x} \cos (bx) \ dx \ (a< \pi)$, you could let $ \displaystyle f(z) = \frac{\mathrm{e}^{(a+ib)z}}{\sinh \pi z} $ and integrate ...
2
votes
1answer
388 views

error of the composite trapezoidal rule

Let $f\in C^2[a,b]$. The composite trapezoidal rule is given by $$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$ First, I've ...
6
votes
7answers
282 views

Looking for an elementary solution of this limit

I was collecting some exercises for my students, and I found this one in a book: compute, if it exists, the limit $$ \lim_{x \to +\infty} \int_x^{2x} \sin \left( \frac{1}{t} \right) \, dt. $$ It seems ...
1
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0answers
58 views

Extremal of functional $ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} $

I have the following functional: $$ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} $$ subject to boundary conditions: $$ \begin{align} y(0) &= 0 \\ ...
0
votes
2answers
56 views

Why is area negative with $\int_0^2 (x^2-3x)dx$

When calculating this integral $$\int_0^2 (x^2-3x)dx = \left [ \frac{x^3}{3}-1.5x^2\right ]_0^2 = \frac{8}{3}-6=-\frac{10}{3}$$ I would use $\frac{10}{3}$ as the result because the area can not be ...
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vote
2answers
152 views

Riemann integral proof

Prove that if the real valued function $f$ on the interval $[a,b]$ is Riemann integrable on $[a,b]$ then so is $|f|$, and $|\int^b_a f(x)dx|\le \int^b_a |f(x)|dx$. The definition I have for Riemann ...
1
vote
3answers
93 views

Wave equation is a well posed problem?

Let $g:\mathbb{R} \rightarrow \mathbb{R}$ and $h:\mathbb{R} \rightarrow \mathbb{R}$ be smooth functions. Prove that the following problem is well-posed: $$ u_{tt}(t,x)=u_{xx}(t,x) \:\:\: t>0, ...
1
vote
0answers
35 views

Integral of a determinant of Jacobian depends on the boundary values only

Let $B$ be the closed unit ball in $\mathrm{R}^n$ with the 2-norm. Let $\phi : B \to \mathrm{R}^n$ be smooth such that $\det D \phi = 1$ on $\partial B$. Why is $\int_B \det D \phi = \int_B 1$? In ...
1
vote
2answers
144 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
0
votes
1answer
74 views

Calculating an integral

Let $r>0$. Show that $$\frac{1}{\sqrt{4\pi t}}\int\limits^{-r}_{-\infty} \exp \left(-\frac{x^{2}}{4t}\right) dx=\frac{1}{2}\left[1-\text{erf}\left(\frac{r}{\sqrt{4t}}\right)\right]$$ where ...
0
votes
2answers
154 views

Computing volume by cross section method.

The base of the solid below is the region in the $xy$-plane bounded by the $x$-axis,the graph of y = $\sqrt{x}$ and the line $ x = 3 $. Find the volume of the solid. Each cross-section of S ...
0
votes
2answers
72 views

Integrate this abstract integral

I want to integrate(in this case: find an antiderivatie) $\int f(\sqrt{c^2+x^2})x\, dx$, where $f \in C$ and $F'=f$ for some $F' \in C^1$ and $c \in \mathbb{R}$. It seems to me as if I'm missing the ...
1
vote
2answers
494 views

Function which is bounded and continuous except on a finite set of points is Riemann Integrable

I am trying to solve the following problem (Problem 7.2.15 of Bartle/Sherbert Book: Introduction to Real Analysis). The problem says: If $f$ is a bounded function on $[a,b]$, and there is a finite ...
2
votes
2answers
111 views

Hints on evaluating this complex integral?

I have the following integral I'm trying to solve: $$\frac{3}{2\pi}\int_0^{2\pi}\frac{e^{-ikx}}{5 - 4\cos(x)} dx, \quad k \in \mathbb{Z}.$$ I've tried writing the exponential in terms of sines and ...