Articles of integration

Evaluate $\int _0^{\infty }\frac{e^{-t^2}-e^{-4t^2}}{t^2}dt$

As my title says, I need help integrating $$\int _0^{\infty }\frac{e^{-t^2}-e^{-4t^2}}{t^2}dt$$ How would approach this? Thanks in advance.

Integral involving Gauss Hypergeometric function, power, exponential and Bessel Function

I am trying to evaluate the following integral involving the Gauss Hypergeometric function, power, exponential and a Bessel Function: $$ \int_0^\infty x e^{-cx^2} {_2F_1(1,\frac{2} {ab},1+\frac{2} {ab},-zx^{-a})I_2(qx)dx} $$ where $a,b,c,z,q > 0 $ real numbers. Can you please provide some hints on solving this integral? Thank you for your time and patience

Prove $(n + 1)I(n) = (n + 2)I(n + 2)$ by integration by parts.

$I_n$ is given by $\int ^ {\pi/2}_{0} \sin^n(x) dx$ My attempt: I got to the fact that the statement is true iff $$\dfrac {n+1}{n+2} = \int ^ {\pi/2}_{0} \dfrac {\sin^2(x)}{\sin^n(x)}$$ I do not know how to prove the above statement using integration by parts.

Evaluate $\int\sin(x^2)\mathrm{d}x$

This question already has an answer here: what is the$ \int \sin (x^2) \, dx$? 3 answers

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 […]