2
votes
2answers
54 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
0
votes
0answers
19 views

Convergence (Absolutely or conditionally) of improper integral.

I need to find out for which values of $\alpha$ the following integral converges (and find out how exactly - absolutely or contidionally): $$ \int\limits_0^{+\infty} x^\alpha \tan\left(\sin\frac ...
7
votes
2answers
195 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
1
vote
2answers
64 views

Improper Integral $\int_{1}^{\infty} \sin \left( \sin\left( \frac {1}{\sqrt{x}+1} \right) \right) dx$

I need to calculate this improper integral. $$\int_{1}^{\infty} \sin \left( \sin\left( \frac {1}{\sqrt{x}+1} \right) \right) dx$$ How do I prove that $$ \sin \left( \sin\left( \frac {1}{\sqrt{x}+1} ...
3
votes
5answers
211 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
0
votes
2answers
55 views

Evaluate the improper integral

$$\int_0^\infty \dfrac{\arctan(ax)-\arctan(bx)}{x}~\mathrm{d}x$$ where $a$ and $b$ are positive real numbers I could not think of a way where to proceed from. Please help!
1
vote
1answer
54 views

Prove that the improper integrals are equal

Prove that $$\int_0^{\infty} \frac{\cos{x}}{1+x} dx = \int_0^{\infty} \frac{\sin{x}}{(1+x)^2} dx$$ Things that I tried so far: I tried to create integral (0, infinity) cos x/1+x - sin x/(1+x)^2 ...
27
votes
3answers
1k views

Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}dx$$
3
votes
2answers
160 views

Solve the following definite integral: $\int_{0}^{\infty}\frac{x^2dx}{({1-x^2})^2}$

Solve the following integral: $$\int_{0}^{∞}\frac{x^2dx}{({1-x^2})^2}$$ I know that substituting some trigonometric functions may help. But I was not able to solve. Can you give me some ...
5
votes
0answers
129 views

Proving $\int_0^1\frac{\log (1-x)}{x}\mathrm dx=-\frac{\pi^2}6$ [duplicate]

It is a well known fact that $\displaystyle\sum_{k=1}^{\infty}\frac1{k^2}=\frac{\pi^2}6$. I wanted to prove this using elementary techniques. By doing some easy algebra, I found it was sufficient to ...
4
votes
0answers
88 views

Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma ...
2
votes
2answers
199 views

$\int_{-\infty}^\infty \frac{\lambda l}{2\pi\epsilon_0(x^2+l^2)^{3/2}}dx$ Proving an electric field from a wire falls off at $1/l$

If you don't know the physics behind all this, that's okay, I just need the integral of this function (or limit, I'm not too sure). Here's the gist: normally with infinitesimal point charges, there ...
35
votes
3answers
817 views

Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$

Prove that $$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
3
votes
2answers
197 views

Some integral with sine

$$\begin{align} & \int_{0}^{+\infty }{\frac{\sin px}{1+{{\text{e}}^{qx}}}}\text{d}x ,\ \ p,\ q>0\\ \\ \\ & \int_{0}^{+\infty }{{{\left( \frac{\sin x}{x} \right)}^{n}}\text{d}x} \\ ...
13
votes
4answers
666 views

$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral

Here is a fun integral I am trying to evaluate: $$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$ I thought about integrating by parts $2n$ times and then using ...