computation of $\int_0^T \frac{\sin (t)}{t} dt$

Question

I need approaches to solve the problem of the following type $\int_{0}^{T} \frac{\sin (t)}{t} dt $. Is there a closed form solution available.

$\frac{\sin x}{x}$ can't be integrated in the sense that it's integral can't be expressed as a combination of a finite number of elementary functions. — Secret Math, Jun 18 '13 at 22:18
@RaM1188 The use of $T$ suggests that this shows up as part of some kind of problem related to periodic functions. If that's the case, I suggest you put all the info in the question. — Git Gud, Jun 18 '13 at 22:24

score 2 · Accepted Answer · answered Jun 18 '13 at 22:51

2

The integral $$ \int_0^T\frac{\sin(t)}{t}\,\mathrm{d}t $$ is called the Sine Integral. It has no closed form in terms of elementary functions.

However, its limit as $T\to\infty$ is $$ \int_0^\infty\frac{\sin(t)}{t}\,\mathrm{d}t=\frac\pi2 $$

answered Jun 18 '13 at 22:51

robjohn

Dear Rob, I had a very old account here. I lost my pass and user name for it and in fact I lost it. Could it be merged with my currently profile? Thanks – Mikasa Jun 19 '13 at 06:03
@BabakS.: I don't see any such account. You should contact the Community Team using the contact us link below. – robjohn Jun 19 '13 at 09:52
I 'll do that. Thanks. May I give you the link indicating that account? – Mikasa Jun 19 '13 at 10:27
If you can't merge using the Merge user profiles page, you will need to contact the Community Team. – robjohn Jun 19 '13 at 11:16

1 Answers1