# integration's questions - Chinese 1answer

42.722 integration questions.

### Multivariable Integral without Fubini

0 answers, 9 views real-analysis integration
I a trying to solve the following problem: Let $K=\{(x,y,z)\in\mathbb{R}^{3} : |x|+|y|\leq 1,\ z\geq 0,\ x^2 +y^2+z^2\leq 1\}$ Show that $\int_K xz^2 dxdydz = 0$ without using Fubini's theorem. I ...

### An integral identity involving Gamma function

0 answers, 19 views integration definite-integrals

### -4 Help in integration problem

Let $y(x)$ be the function $$y(x)=\begin{cases}10 &\text{if} \ x=5 \\ 0 &\text{otherwise}\end{cases}.$$ Calculate the integral $$\int_{-\infty}^{\infty} y \ dx,$$ Is the answer 10? Please ...

### Wrongly calculated integral

Calculate $$\iiint_V \frac{1}{(x+y+z)^3} \, dV$$ Where $V$ is the volume bounded by the planes $$\{4x + 3z = 12, 4x + z = 4, 4y + 3z = 12, 4y + z = 4, z = 0\}$$ I've used simple coordinates ...

### What do you need to know to do the following integral

Integrate $$\int_{-\infty}^{\infty} \cos(\sqrt{2} x)e^{-x^2} dx$$ All i want to know is how to proceed. I have been stuck at this for a very long time now. I tried writing the $\cos$ term as ...

### 4 Closed-form solution for an integral with integer power rational function

3 answers, 57 views calculus integration
I need a closed-form solution for the following integral: $$\int_{0}^{\infty}\frac{dt}{(t+a)^{m} (t +b)^{n}};\,\,a,b>0;\,\,m,n\geq1\,\text{are integers}$$ If $m$ and $n$ are non-integers, I can ...

### 3 Why do we ignore constant produced by integration of derivative when we derive integration by parts formula?

2 answers, 39 views calculus integration
My question is somehow related to this question. What I would like to know is why we ignore the fact that integral of derivative of f(x) is equal to ...

### substitution for a multiple integral

1 answers, 33 views integration substitution jacobian
I came across a concept which I have never dealt with before. I had an idea but it is flawed and I would like to know why it is flawed. To those who are familiar with that subject it will seem ...

### 1 generalized Riemann Integrability of $f\cdot g$

Let $\mathcal{R}([a.b])$ the set of all Riemann-integrable functions in $[a,b]$. Let $\mathcal{R}^{*}([a,b])$ the set of all Generalized Riemann-Integrable functions in $[a,b]$ (I'm talking about the ...

### 2 $\lim_{\epsilon\rightarrow 0^{+}}{\int_{0}^{\infty}\,e^{-\epsilon x}\,\frac{|\sin{x}|^{t}} {x^{t-1}}\,dx}, \quad 1< t<2$

let $1<t<2$. I need to evaluate $$\lim_{\epsilon\rightarrow 0^{+}}\int_{0}^{\infty}\,e^{-\epsilon x}\,\frac{|\sin{x}|^{t}} {x^{t-1}}\,dx$$ If $t>2$ one can easily apply the dominated ...

### 38 Is there a chain rule for integration?

6 answers, 79.818 views calculus integration
I know the chain rule for derivatives. The way as I apply it, is to get rid of specific 'bits' of a complex equation in stages, i.e I will derive the $5$th root first in the equation $(2x+3)^5$ and ...

### 20 Class of integrals: $I(a)=\int_0^\infty \frac{dx}{e^x+ax}$

I'm investigating integrals in the form $$I(a):=\int_0^\infty \frac{dx}{e^x+ax}$$ So far, I haven't been able to find any special values other than $I(0)=1$, and I've only managed to evaluate these ...

### Definition of a positively integrable function

1 answers, 19 views integration lebesgue-integral