Articles of integration

Finding the antiderivative of the product of two functions given only their derivative properties

let $\alpha'(x)=\beta(x), \beta'(x)=\alpha(x)$ and assume that $\alpha^2 – \beta^2 = 1$. how would I go about calculating the following anti derivative : $\int (\alpha (x))^5 (\beta(x))^4$d$x$. Thank you.

Evaluating $ \int \frac{1}{5 + 3 \sin(x)} ~ \mathrm{d}{x} $.

What is the integral of: $\int \frac{1}{5+3\sin x}dx$ My attempt: Using: $\tan \frac x 2=t$, $\sin x = \frac {2t}{1+t^2}$, $dx=\frac {2dt}{1+t^2}$ we have: $\int \frac{1}{5+3\sin x}dx= 2\int \frac 1 {5t^2+6t+5}dt $ I’ll expand the denominator: $5t^2+6t+5=5((t+\frac 3 5 )^2+1-\frac 1 4 \cdot (\frac 6 5)^2)=5((t+\frac 3 5)^2+0.64)$. So: $2\int \frac 1 {5t^2+6t+5}dt = \frac […]

Solving a non-linear Partial differential equation $px^5-4q^3x^2+6x^2z-2=0$

I have to find out the complete integral of : $px^5-4q^3x^2+6x^2z-2=0$ My attempt:Let $f(x,y,z,p,q)=px^5-4q^3x^2+6x^2z-2$ So, $f_p=x^5,f_q=-12q^2x^2,f_x=5px^4-8q^3x+12xz,f_y=0,f_z=6x^2$ Applying Charpit’s method: $\frac{dx}{x^5}=\frac{dy}{-12q^2x^2}=\frac{dz}{px^5-12q^3x^2}=\frac{-dp}{5px^4-8q^3x+12xz+6px^2}=\frac{dq}{12x^2q^2} \implies \frac{dy}{-12q^2x^2}=\frac{dq}{12x^2q^2}\implies q=\sqrt{(y+c)}$ where $c$ is an arbitrary constant. I don’t see here a way to find out $p$. Please help me solve this problem .Even a hint would be much help. I’ll appreciate any […]

Find by integrating the area of ​​the triangle vertices $(5,1), (1,3)\;\text{and}\;(-1,-2)$

Find by integrating the area of ​​the triangle vertices $$(5,1), (1,3)\;\text{and}\;(-1,-2)$$ I tried to make straight and integrate, but it is very complicated, there is some better way?

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: $$\mathcal{S}:=\{s:\Omega\to\mathbb{C}:s=\sum_{k=1}^{K<\infty}s_k\chi_{A_k:\lambda(A_k)<\infty}\}$$ Denote the positive and negative part of the real and imaginary part by: $$f=\Re_+f-\Re_-f+i\Im_+f-i\Im_-f=:\sum_{\alpha=0\ldots3}i^\alpha f_\alpha$$ Define for positive functions: $$\int fd\lambda:=\sup_{s\in\mathcal{S}:s\leq f}\int sd\lambda\quad(f\geq 0)$$ and for complex functions: $$\int fd\lambda:=\sum_{\alpha=1\ldots3}i^\alpha\int f_\alpha d\alpha$$ as long as all terms of […]

Are the Euler substitutions valid only for rational functions?

I am self-studying, Now I try to learn integrals, Especially the Euler’s substitution. if $a > 0$: we put $\sqrt{ax^2+bx+c} = \pm \sqrt{a}x+t$ if $c \ge 0$ : $\sqrt{ax^2+bx+c} = xt \pm \sqrt{c}$ In books, I find that these substitutions are applicables for the following form: $$ \int{R(x, \sqrt{ax^2+bx+c}) \mathrm{d}x} $$ My question is: are […]

About polar coordinates in high dimensions

I’m trying to understand a proof in Michel Willem, Functional Analysis — Fundamentals and Applications, Birkhäuser. The book defines: $$\int_{\Bbb S^{N-1}}f(\sigma)\,d\sigma=N\int_{B_N}f\left(\frac{x}{|x|}\right)dx.$$ And then goes on to proving: $\textbf{Lemma 2.4.7.}$ Let $u\in{\cal K}(\Bbb R^N)$. Then (a) for every $r>0$, the function $\sigma\mapsto u(r\sigma)$ belongs to $C(\Bbb S^{N-1})$; (b) $\displaystyle\frac{d}{dr}\int_{|x|<r}u(x)\, dx=r^{N-1}\int_{\Bbb S^{N-1}} u(r\sigma)\,d\sigma;$ (c) $\displaystyle \int_{\Bbb R^N}u(x)\, […]

Show that for $|f_n| \le g_n$ $\forall n$: $\lim_{n\to \infty} {\int_E g_n } = \int_E g \Rightarrow \lim_{n\to \infty} {\int_E f_n } = \int_E f$

Let $(f_n)_{n \in \Bbb N}$ be a series of measurable functions on E, that converges almost everywhere pointwise towards $f$. Let $(g_n)_{n \in \Bbb N}$ be a series of on $E$ integrable functions that converge almost everywhere on $E$ pointwise towards $g$. Also, $|f_n| \le g_n$ $\forall n \in \Bbb N$. I have to show […]

Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann Sum and Upper Riemann sum are bounded .More mathematically $m(b-a) \leq L(P,f) \leq U(P,f) \leq M(b-a)$ where $m,M$ are lower and upper bounds of the function $f$ respectively. Rudin […]

Solution to differential equation $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$

I’m trying to solve the following differential equation: $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$ here $g(x,y,z+h)$ is a known function that however i will leave unspecified moreover we are dealing with Real variables and constants. The paper that i’m following reports the following solution: $w(x,y,z)= \frac{1}{2\lambda}\int_{0}^{\infty}e^{-\frac{s}{2\lambda}}g(x,y,z+h+s)ds$ I’m sure this comes from an exponential integrating factor, in fact arranging […]