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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1answer
738 views

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

Again, I am new to volume of bodies and I am struggling with it. Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$. Now from a previous question, I know that I can do it by ...
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2answers
39 views

How to evaluate base indefinite integrals

I can find a list of known integrals anywhere, but how would I build this list myself? For example how can I prove that $\int x^2 dx = x^3 / 3$? I want to understand the general theory. It would be ...
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2answers
282 views

Find k such that f is density function

I have the following function: $f_X(x, \theta) = \left\{ \begin{array}{lr} k/x^3 & : x \leq \theta \\ 0 & : x > \theta \end{array} \right.$ and $\theta >0$. I ...
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1answer
42 views

Approximate an Integration by a linear formula

I just wonder are there any methods to approximate the following integration by a linear formula ? $$ \int_{x_1}^{x_2} \int_{y_1}^{y_2} f( x,y,w_1,\dots,w_n ) \, dx \, dy \approx \sum\limits_{i = ...
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1answer
43 views

Problem with understanding integration by parts method?

First of all (before I get started with integration by parts method to solve integrals) I need to know what is the meaning of all terms in integration rules like $a^x,\,e^x,\,\ln\left(x\right),\,$ ...
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3answers
284 views

Evaluating $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ using polar coordinates

Use polar coordinates to evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ I understand that we need to change $x^2+y^2$ to $r^2$ and then we get $\int_0^1 \int_0^{\sqrt{1-x^2}} ...
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1answer
62 views

Evaluating: $\int^{n}_{1}[\ln(x) - \ln(\lfloor x \rfloor)] dx $

I am attempting to evaluate the integral: $$\int^{n}_{1}\ln(x) - \ln(\lfloor x \rfloor) dx $$ To a form: $$f(x) + O(g(x))$$ where $g(x) \rightarrow 0$ as $x \rightarrow \infty $ How do I compute ...
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2answers
34 views

What´s the criteria for choose a appropriate in this question?

Taking a look at this page, Why did the editor choose $\ln(x)$ for $u$. It´s because is more simple to differentiate?
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0answers
91 views

Intersection of a hollow cone and a circular disc in 3D

I'm trying to calculate the area of a hollow cone intersecting a circular region on a surface. Basically a hollow cone is defined by its starting apex and its ending apex, the height of the cone from ...
3
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2answers
110 views

Wolfram double solution to $\int{x \cdot \sin^2(x) dx}$

I calculated this integral : $$\int{x \cdot \sin^2(x) dx}$$ By parts, knowing that $\int{\sin^2(x) dx} = \frac{1}{2} \cdot x - \frac{1}{4} \cdot \sin(2x) +c$. So I can consider $\sin^2(x)$ a ...
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3answers
162 views

I'm looking for several ways to prove that $\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$

I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$$ for $-2< Re(m)< 0$
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2answers
167 views

Find the volume of the body bounded by $x^2 + y^2 + z^2 = 4$ and $x^2 + y^2 = 1$

Find the volume of the body bounded by $x^2 + y^2 + z^2 = 4$ and $x^2 + y^2 = 1$. This is the last subject in our syllabus and I am afraid that my professor had not any time to teach it before ...
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2answers
78 views

integration by parts to solve this problem

Can someone please show me the steps to solve $$ \frac{\sqrt{5}}{2}\int_{-1}^{1}xe^{-\frac{1}{2}x+\frac{3}{2}}dx $$ with integration by parts. I did it twice and got all different answers. I know the ...
4
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1answer
202 views

integral involving cot coth and e using contour

I am thinking it is possible to evaluate this integral using contours, but I got hung up. $\displaystyle \int_{0}^{\infty}e^{-ax}\sin(ax)\left(\cot(x)+\coth(x)\right)dx=\frac{\pi}{2}\cdot ...
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2answers
144 views

Analytic solution of $ \int_t^{t+r} \frac{x^2}{(t+r-\sqrt{2rx-x^2})^4}dx $

Could anyone help me to solve this defined integral analytically? $$ \int_t^{t+r} \dfrac{x^2}{(t+r-\sqrt{2rx-x^2})^4}dx $$
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2answers
56 views

Relations between different types of integrals

There are many types of integrals: Riemann, Lebesgue (and, more generally, integral by measure), integral of differential forms, etc. What are relations between them? When they are agree and when ...
3
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2answers
239 views

What went wrong? Calculate mass given the density function

Calculate the mass: $$D = \{1 \leq x^2 + y^2 \leq 4 , y \leq 0\},\quad p(x,y) = y^2.$$ So I said: $M = \iint_{D} {y^2 dxdy} = [\text{polar coordinates}] = \int_{\pi}^{2\pi}d\theta ...
2
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2answers
145 views

Integrating Factorials

I feel like I'm doing something wrong here: $$\frac{d^n}{dx^n}(x^n)=n!$$ $$ 5!=\frac{d^5}{dx^5}(x^5)$$ $$ \int{5! dx}=\int{\frac{d^5}{dx^5}(x^5)}dx=x\frac{d^4}{dx^4}(x^4)=x*4!$$ Please explain what ...
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4answers
497 views

Partial fractions to integrate$\int \frac{4x^2 -20}{(2x+5)^3}dx$

$$\int \frac{4x^2 -20}{(2x+5)^3}dx$$ I can't use the coverup method that I learned since making anything zero in this makes everything zero. I would probably just use random test points because I ...
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2answers
54 views

Please help to solve this ODE with function coefficients

Is it possible to solve this ODE for $y$? According to wikipedia this falls in the category of a first order, linear, inhomogeneous ODE with function coefficients. But is there a more tractable ...
3
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1answer
86 views

$u$-substitution for integrating $\int\frac{\log|x|}{x\sqrt{1+\log|x|}}\,dx\;\;?$

How can I integrate $$\int\frac{\log|x|}{x\sqrt{1+\log|x|}}\,dx\;\;?$$ I'm not sure what I should put equal to $\,u.$ Can someone give me a hint on how to solve this question? I don't need a full ...
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3answers
88 views

Integrating $\int^{e^3-1}_{0}\frac{dt}{1+t}.$

How can I integrate $$\int^{e^3-1}_{0}\frac{dt}{1+t}.$$ I tried to make $u=1+t$ which means that $du=dt$ but it's not giving me anything useful, but instead made things more complicated. Maybe I did ...
5
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2answers
83 views

Using $\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! n^z}{z(z+1)(z+2)\ldots(z+n)} \right]$ to prove the Weiestrass product.

I was searching the web quite thoroughly in the last two days. I was in paralytically looking for a rigorous proof using $$ \Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! ...
3
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4answers
295 views

Partial fraction integration $\int \frac{dx}{(x-1)^2 (x-2)^2}$

$$\int \frac{dx}{(x-1)^2 (x-2)^2} = \int \frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}+\frac{D}{(x-2)^2}\,dx$$ I use the cover up method to find that B = 1 and so is C. From here I know that the ...
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4answers
155 views

Example of integral pairs or triples ($I$, $J$, $K$…)

A somewhat common trick when evaluating integrals is to add a second (or third) integral to help integrate the first. As an example if one where to compute $$ I = \int \sin \log x \, ...
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0answers
20 views

Applying continuous operators to functions defined by parameter integrals.

Let $T\in \mathcal{L}(\mathcal{S}(\mathbb{R}^{n}))$. For fixed $f\in\mathcal{S}(\mathbb{R}^{2n})$, define $g(x):=\int_{\mathbb{R}^n} f(x,y)\, dy$. Note that this implies that $g$ is also Schwartz. ...
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4answers
172 views

Substitution for $\int \frac {dx} {ax^2 + bx + c}$

I'm looking for the substitution that makes easier to solve integral containing quadratic polynomial in denominator (!) when such polynomial cannot be broken into parts (if it can, then it's possible ...
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1answer
52 views

About divergence theorem

Consider the portion $S$ of the sphere $x^2+y^2+z^2=4$ with $z\ge -1$. Calculate the integral $$\iint_{S} (x^3, y^3, z^3)\cdot \vec{n} dS$$ a) Using directly a parametrization Well, what are the ...
3
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1answer
115 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
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4answers
309 views

Evaluating $\int \frac{dt}{(\cos(t))^2}$?

How do I solve an integral with a differential on top? E.g.: given this integral to evaluate: $$\;\int \frac{dt}{(\cos(t))^2}\;\;?$$ What does it even mean when there's a differential?
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0answers
50 views

About the functional inverse of integrals and infinite products.

It seems $\cos(x)$ and $\sin(x)$ are the only entire functions, that are the functional inverse of an integral of some elementary function $f(x)$ , such that they have a simple infinite product ...
2
votes
3answers
90 views

Partial fraction $\int \frac{x^2 + 11x dx}{(x-1)(x+1)^2}$

I have been using the cover up method from this video lecture $$\int \frac{x^2 + 11x}{(x-1)(x+1)^2} dx$$ $$\frac{x^2 + 11x }{(x-1)(x+1)^2} = \frac{A}{x+1} + \frac{B}{x-1} + \frac{C}{x-1}$$ What do ...
10
votes
1answer
322 views

Writing Integrals using Differential Forms

Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
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votes
5answers
460 views

The indefinite integral $\int{\frac{\mathrm dx}{\sqrt{1+x^2}}}$

I need to solve this integral: $$\int{\frac{\mathrm dx}{\sqrt{1+x^2}}}$$ First I thought it was easy, so I tried integration by parts with $g(x)=x$ and $g'(x)=1$: $$\int{ ...
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votes
2answers
192 views

Using trig substitution to evaluate $\int \frac{dt}{( t^2 + 9)^2}$

$$\int \frac{\mathrm{d}t}{( t^2 + 9)^2} = \frac {1}{81} \int \frac{\mathrm{d}t}{\left( \frac{t^2}{9} + 1\right)^2}$$ $t = 3\tan\theta\;\implies \; dt = 3 \sec^2 \theta \, \mathrm{d}\theta$ $$\frac ...
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3answers
50 views

First oder linear differential equation, wrong result

I have a differential equation: $y'= \frac{y}{x} + x^2$ I apply this formula: $y(x)= e^{A(x)} \int {e^{-A(x)} b(x) dx} $ With $a(x) = \frac{1}{x}$ and $b(x)= x^2$, and $A(x)$ primitive of ...
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1answer
80 views

Interchanging limit and integration when bounds of integration depend on a parameter

This is yet another question on when it is permissible to interchange limits and integrals. I am interested in the situation when bounds of integration depend on some parameter, and then the limit is ...
5
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2answers
186 views

Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$

$$\int \frac{dx}{\sqrt{x^2 -9}}$$ $x = 3 \sec \theta \implies dx = 3 \sec\theta \tan\theta d\theta$ $$\begin{align} \int \frac{dx}{\sqrt{x^2 -9}} & = \frac{1}{3}\int \frac{3 \sec\theta ...
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1answer
31 views

Continuity of an integral with respect to one variable

Let $V\subseteq \mathbb{R}^n$ and $f:V\to\mathbb{R}^n$. Consider the function $$g(x_1,x_2,...,x_n) = \int_{x_2}^{x_1} {f(t,x_2,...,x_n)dt}$$ on $V$. What conditions will I need to conclude that $g$ is ...
2
votes
2answers
142 views

Proving The Average Value of a Function with Infinite Length

This is the given: One can extend the definition of the average value of a continuous function $f(x)$ to the interval $[a,\infty)$ of infinite length as follows: ...
2
votes
2answers
47 views

Comparing Areas under Curves

I remembered back in high school AP Calculus class, we're taught that for a series: $$\int^\infty_1\frac{1}{x^n}dx:n\in\mathbb{R}_{\geq2}\implies\text{The integral converges.}$$ Now, let's compare ...
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2answers
123 views

Fundamental Theorem of Calculus in Multivariate Case

From the FTC we have, for continuously differentiable $f: \mathbb{R} \to \mathbb{R}$, $$ f(a) - f(b) = \int_b^a \frac{d}{dx} f(x) dx $$ I'm trying to write the difference between a vector function ...
2
votes
2answers
1k views

Find the centroid bounded by the curves of $y+x=2$, $y^2=x$

Problem Statement: Find the centroid of $y+x=2, y^2=x$ I think my main problem is finding the limits of integration. I originally set it at 0 to 1, but that didn't work.
5
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1answer
94 views

Is this function bounded? Next question about integral $\int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac1{||y-x||} dS_y$.

Let $\partial M$ be $C^2$ closed surface in $\mathbb{R}^3$, $M$ is open. Show that $$ f(x) = \frac{\int_{\partial M} \left| \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{1}{||y-x||} \right| ...
2
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1answer
94 views

Inequality involving definite integral

Just wondering, what may be the best way to show that $$\int_0^1 xf(x)dx \leq \frac{1}{2}\int_0^1 f(x)dx,$$ provided that $f(x) \geq 0$ over the interval $[0,1]$ and that $f(x)$ is monotonically ...
2
votes
2answers
277 views

Riemann-Stieltjes integral (floor function)

Please, I am having problem with this function. $$ \int_{-1.2}^{3.9} xd[x] $$ Here is what I have done $$\int_{-1.2}^{1}xd[x] + \int_{0.8}^{2}xd[x] + \int_{1.8}^{3}xd[x] + \int_{2.8}^{3.9}xd[x] $$ ...
0
votes
2answers
156 views

Conditional/Absolute convergence of $\int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$

I need to check conditional/absolute convergence of the integral: $$f(x) = \int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$$ I tried for a long time and I can't understand what I should do. I know that ...
5
votes
0answers
268 views

Prove the equation: $\frac{2}{\pi} \int_0^\infty \frac{\cos kr - ak \sin kr}{k^2a^2 +1} \ldots $

Prove the following equation: \begin{equation} \frac{2}{\pi} \int_0^\infty \frac{\cos kr - ak \sin kr}{k^2a^2 +1} \left (\int_0^\infty \cos kr' \left [u(r')-au'(r') \right] dr' \right ) dk =u(r) . ...
0
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0answers
21 views

Properties of integrating characters, traces

Here are two steps from proofs about traces in Lang's $SL_2$ (pp. 30, 31) that I didn't understand. Let $\pi$ and $\sigma$ be representations of $G = GL_2(\mathbb R)$ or $SL_2(\mathbb R)$ in $H_\pi$, ...
23
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1answer
509 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...