Articles of integration

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions and one particular solution of the homogeneous equation:

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions of this equation: $y_2=\ln(x)$ $y_1=x+\ln(x)$ and one particular solution of the homogeneous equation: $y_3=e^{2x}$ I think we can take two linear independent solutions: $x$ and $\ln(x)$ make $Y=c_1x+c_2\ln(x)+e^{2x}$ but i don’t know what to do next

Complex analysis.Line integrals

Show that the $\mid\int(e^z-\bar{z})\mid$ over the positive oriented triangle γ with end points $z = 0$ $ z = 3i$ and $z = −4$ is less than 60.$$\mid\int(e^z-\bar{z})\mid<60$$ Now i know on my curve $Re(z)<0$ and $\mid{z}\mid$<4 or equal . So $\mid\int(e^z-\bar{z})\mid \leq \int\mid(e^z-\bar{z})\mid) \leq \int\mid(e^z\mid+\mid\bar{z})\mid =\int e^{Re(z)}+\mid z \mid \leq \int e^0+4=5\int dz$ Now […]

How to calculate a limit of this function for checking integral convergence?

$$\lim _{x\to 1^+}\frac{\ln x}{\sqrt{x^3-x^2-x+1}}$$ I want to calculate the check if the integral $$\int _2^{\infty }\frac{\ln x}{\sqrt{x^3-x^2-x+1}}$$ is converge. I want to do so with the comparison test.

Show that $f(x)\equiv 0$ if $ \int_0^1x^nf(x)\,dx=0$

Let $f \in C[0,1]$. If for each integer $n \ge 0$ we have $$ \int_0^1x^nf(x)\,dx=0$$ show that $f(x) \equiv 0$

A Property of Integrals defined for step functions.

This question is an exact duplicate of: Additive Property of Integrals of Step Functions [duplicate] 2 answers

Derivatives and Integrals of Polynomials and more.

I noticed that if I had a function $f(x)=x^n$ where $n$ is an integer, then $\lim_{m\to{n^+}}f^{(m)}(x)=n!$ where $f^{(m)}(x)$ is the $m$-th derivative. Also, $$\lim_{m\to{n^-}}f^{(m)}(x)=\frac{(-1)^n\ln(x)}{n!}$$Where we have $m,n$ as integer numbers. The limits I use in this question are not limits of the normal sense. Instead, they are used to describe positives, negatives, and where it […]

Multivariate Gaussian in a Sphere

Suppose that $X \in \mathbb{R}^n$ is a multivariate normal with a covariance given by the identity $I$ . Also, let $S_r$ be an $n$- dimensional sphere with radius $r$. How to compute $Pr[ X\in S_r]$? In my attempt \begin{align} Pr[ X\in S_r]=\int_{S_r} \frac{1}{\sqrt{(2 \pi)^n n}} e^{-\frac{1}{2} x^T x} dx = \int_{S_r} \prod_{i=1}^n \frac{1}{\sqrt{2 \pi}} e^{-\frac{x_i^2}{2} […]

Getting rid of square root via integration

How do we prove, for positive $D$, this result? $$ e^{-2\sqrt D} \sqrt{\pi} = \int_0^\infty s^{-1/2} e^{-(s+D/s)} ds $$

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven’t been able to figure out the solution myself. $$ \int\limits_{(n-1)T}^{nT}\left[\frac{2\pi}{T}\displaystyle\sum_{i=2}^{\infty}\left(\frac{TK}{2\pi}f(x)\right)^i\right]dx $$ given that: $T$ and $K$ are constants $ \int\limits_{(n-1)T}^{nT}Kf(x)dx = y[n] $ $ f(x) $ does not change significantly between $ (n-1)T […]

How to solve $\int_0^{\infty}\frac{x^n}{(x^2+1)^n}\,\mathrm dx$ for $n\ge 2$?

I am trying to find some closed form answer for the integral $$\int_0^{\infty}\frac{x^n}{(x^2+1)^n}\,\mathrm dx,\quad n\ge 2$$ I am not sure if a closed form exists and I have been trying this integral for hours. Any tips or hints would be appreciated.