# Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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### Calculate $\int_{0}^{1} \int_{y}^{1} e^{-x^2} \, \mathrm{d} x \mathrm{d}y$

Calculate $\displaystyle \int_{0}^{1} \int_{y}^{1} e^{-x^2} \, \mathrm{d} x \mathrm{d}y$ I understand that multiple integral has to be used, but I can't go further. Can anyone show me how to solve ...
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### Find the Area under $R$ Using Integrals

$R$ is a region in the first octant of $\mathbb{R^3}$ that is bounded by $x^2+y^2+z^2=4$ and $z=\sqrt{x^2+y^2}$. Set up the integral for the volume of this region $R$. I think the first thing to do is ...
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### Solution to $\int\frac{1}{x+xe^{ax+b}}dx$

I need to solve $\int\frac{1}{x+xe^{ax+b}}dx$ as part of a larger optimization problem, where the $a$ and $b$ parameters will be optimized in a subsequent step. Wolfram alpha returns the following ...
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### Help with Integral Involving fractions and exponentials

Hi I need help with the integral below. I have had a tried a few substitutions but they don't work. $\int_{z_u}^{z_f} \frac{dz}{\sqrt{\frac{5+\gamma e^{-bz}}{\delta^2}-6}}$ where $\delta$, $\gamma$ $a$...
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### Is there an antiderivative for $e^{-\left( x + \frac{1}{x}\right)}$?

I was playing around with some integrals I made up myself and was trying to find a closed-form for $$\int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx, \qquad t < \infty$$ I'm aware that if ...
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### Show Riemann-integrability of $h:=\max\{f,g\}$

Let be $f,g$ two Riemann-integrable functions on the interval $[a,b]$ where $a<b$. Show that $h:=\max\{f,g\}$ is also Riemann-integrable on $[a,b]$. My approach: First, a bit of notation: \begin{...
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### Theorem 10.2 Rudin

$\mathscr b(X)$ denotes the set of all complex-valued, continuous, bounded functions with domain $X$. I don't understand why is $L(h)$ equal of $\prod_{i=1}^k$ $\int_{a_i}^{b_i} h_i(x_i)dx_i$ and ...
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### Baby Rudin 10.1

I want to be sure that I understand it correctly. As I understand it $\int_{I^k} f(x) dx$ = $\int_{a_{k-1}}^{b_{k-1}}$ ($\int_{a_k}^{b_k} f_k(x_1,...x_{k-1},x_k)dx_k$)$dx_{k-1}$ and to continue this ...
Denote $$\omega_0(t)=\frac{\text{d}t}{t} , \omega_1(t)=\frac{\text{d}t}{1-t}, \omega_2(t)=\frac{\text{d}t}{1+t}.$$ I want to compute($\text{Li}$ denotes polylogarithms) $$I=\int_{0}^{1}\operatorname{... 0answers 60 views ### Help proving that signed volume of n-parallelepiped is multilinear Overview I am trying to build some intuition about the volumes of parallelepipeds and determinants. I would like to define the determinant as the unique function of N vectors in \mathbb{R}^N which ... 1answer 69 views ### Evaluate \iint_D xy [closed] My question is, evaluate the area under the curve, \iint_D xy, where D is a region bounded by y=\sin(x) and y=\cos(x). Here is an illustration of it: As you can see, D is the area between ... 1answer 115 views ### Where I made mistake(s) as the integrating of ~\int_{}^{}\frac{1}{\sin^{2}\left(x\right)}\,dx~?$$L:=\int_{}^{}\frac{1}{\sin^{2}\left(x\right)}\,dx\tag{1}=\int_{}^{}\sin^{-2}\left(x\right)\,dx\tag{2}=\frac{\sin\left(x\right)^{-1}}{\left(-1\right)\left(\cos^{}\left(x\right)\right)}+\...
Is there a function similar to the one in the image that has a discontinuity in the origin, for which: $$\lim_{x \to \pm\infty} f(x)=0$$ and $$\lim_{x \to 0^{\pm}} f(x)=+\infty$$ and \int_{-\infty}^{...