Sine integral Definition and 13 Threads

Hello, I've recently discovered the sine integral and have been playing around with it a bit on some graphing software. I looked at the graph of ##Si(x^2) - \frac π 2## and saw that both the amplitude and period was decreasing as x increased. Curiosity got the best of me so I decided to...

Properly speaking, since sin(x) and cos(x) don't go to zero as x \rightarrow \infty, the following integrals are undefined: \int_0^{\infty} cos(kx) dk \int_0^{\infty} sin(kx) dk However, in the handwavy way of physicists, we can often pretend that the cosine integral "converges" to \delta(x)...

Hello PF, I just found a curious integral. I wondered if it comes from a bigger group of integral definitions: \int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a) Where Si(x) is the sine integral function \mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x I proved the...

Hello, and thanks for welcoming me in the forum of Physics Forums. I just found a curious integral that I solved by Taylor series. I wondered if it comes from a bigger group of integral definitions: ##\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)## Where Si(x) is the sine...

Homework Statement Solve by variation of parameters: y" + 3y' + 2y = sinex Homework Equations Finding the complimentary yields: yc = c1e-x + c2e-2x The Attempt at a Solution I set up the Wronskians and got: μ1 = ∫e-2xsin(ex)dx μ2 = -∫e-xsin(ex)dx The problem is that I have no idea how to...

This thread will be dedicated to find a general formula for the integral $$I(a,t) = \int^t_0 x \log|\sin(a x )| \, dx \,\,\,\,\, a,t>0$$ This is not a tutorial . Any comments or attempts are always be welcomed .

Homework Statement Any ideas for how to solve the following integral? $$\int_{0}^{\pi}\sin{n x}\sin{x}^3 dx$$ where n is a positive integer Homework Equations Various sine and cosine identities The Attempt at a Solution I haven't much of a clue how to solve the integral...

This is not really a homework, I am trying to expand Si(x) into a series.The series expansion of Si(x) is given in articles: Si(x)=\int_0^x \frac{\sin\theta}{\theta}d\theta=\sum_0^{\infty}\frac {(-1)^k x^{2k+1}}{(2k+1)(2k+1)!} This is my work, I just cannot get the right answer: Si(x)=\int_0^x...

Homework Statement Hi. I need help understanding a task where i am supposed to prove that a function must be greater than 0 when x is from 0 and up. f(x) = (x-0)integral of (sinx/(x+1)) please help me out with this. Mons

Homework Statement Si(x)=\int(sint/t)dt from 0 to x integrand is 1 at t=0 express the solutiony(x) of the initial value problem x^3y'+2x^2y=10sinx, y(0)=0 in terms of Si(x) Homework Equations The Attempt at a Solution y'+2/xy=10x^-3sinx multiply by x^2 x^2y'+2xy=10x^-2sinx...

How might I find the integral solution for the following: (Sorry for the poor notation) \intsinc(4x*\sqrt{(1-2/x)}) dx from -\infty to \infty

Homework Statement Integral((sin(x))^2((cos(x))^2) dx Homework Equations The Attempt at a Solution Seperate Cos (x)^3 sin(x)^2 (cos(x))(cos(x))^3 Then:apply identities sin(x)^2(cos(x))(1-sin(x)^2) And now I am lost!:eek: Thanks Alot!

i know that the sine integral converges to pi/2. But what about the abs value of the sine integral. It seems to me that it would have value oo. But I'm having trouble coming up with a lower bound that diverges.