2
votes
0answers
78 views

About differentiation under the integral sign

I would like to ask something related to the application of the differentiation under the integral sign (Leibniz rule) given by ...
3
votes
2answers
111 views

Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$ \rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...
5
votes
1answer
74 views

Differentiation under integral sign help

Question is: If $$f(a)= \int_0^\infty e^{-t^2}\cdot \cos(at)~dt$$ then I have to show that $f'(a)=-\dfrac{a}{2}\cdot f(a)$. I know that ...
1
vote
0answers
62 views

What is the derivation of an integral?

Given an intergral $$ \int_{a(t)}^{b(t)} f(t, x) dx $$ What is the fomula to of its derivation? $$ \frac{\partial}{\partial t} \int_{a(t)}^{b(t)} f(t, x) dx $$ Would someone please give me a link to ...
2
votes
2answers
221 views

$\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx$

Prove that: $\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx=\int_{-\infty}^{\infty} -2x e^{-x^2} \sin(2tx) dx$ This is my proof: $\forall \ t \in \mathbb{R}$ (the improper integral ...
5
votes
2answers
204 views

Differentiating under the integral sign problem

Knowing that $$\int_0^\infty e^{-x^2}\,dx = \frac{\sqrt{\pi}}{2},$$ evaluate the integral $$\int_0^\infty e^{-x^2y+1}\,dx.$$ for $y > 0$
0
votes
1answer
62 views

what can we say about $G(.)$?

Given $c \in R$, a deterministic probability density $f(x)$ and its cumulative distribution $F(c)$, what can be said about $G(c)$ where: $G(c)=\int f(x)F\left( x+c\right) dx $ The question ...