Integrating (sin(x))/x dx -- The limits are a=0 and b=infinitity

[notamathgod]

Homework Statement

I tired integration by parts but it leads me nowhere. How do you solve this using methods of integration that you learn early on. (U-sub - importer integrals)?

Relevant Equations

is this integral possible to solve without using feynamns technique?

U= 1/x dV= sin(x)
dU = -1/x^2dx V= -cos(x). lim b--> infiniti (integral from [0,b]) = 1/x(-cos(x)) - integral(1/x^2(cos(x)) dx

 

[Mark44]

U= 1/x dV= sin(x)
dU = -1/x^2dx V= -cos(x). lim b--> infiniti (integral from [0,b]) = 1/x(-cos(x)) - integral(1/x^2(cos(x)) dx

You might try this (which I haven't worked through):
Working with an indefinite integral, use integration by parts twice, the first time with u = sin(x), dv = dx/x. The second time through pick u = cos(x).
You might end up with the same integral you started with, in which case you can solve algebraically for the integral. If this works, then use limits to evaluate the improper integral.

Again, I haven't worked it through, but that's the tack I would start with.

BTW, "infiniti" is a word cooked up by the marketing weasels at a car company. The word you want is infinity, with a 'y'.

 

[Ray Vickson]

You might try this (which I haven't worked through):
Working with an indefinite integral, use integration by parts twice, the first time with u = sin(x), dv = dx/x. The second time through pick u = cos(x).
You might end up with the same integral you started with, in which case you can solve algebraically for the integral. If this works, then use limits to evaluate the improper integral.

Again, I haven't worked it through, but that's the tack I would start with.

BTW, "infiniti" is a word cooked up by the marketing weasels at a car company. The word you want is infinity, with a 'y'.

All the standard methods are inapplicable: the integration
$$S(x) = \int \frac{\sin(x)}{x} \, dx$$ is provably non-elementary. That means that it is provably impossible to write down the indefinite integral as a finite expression in "elementary" functions. Even if you allow for a 100-billion page formula, that would not be enough to express the integration. (On the other hand, a simple expression involving an infinite series can be given for $S(x)$, but an infinite series is not, formally, a finite expression.)

However, certain special cases are do-able---not by finding antiderivatives, but by using other methods such as contour integration. For example, the definite integral
$$\int_0^\infty \frac{\sin(x)}{x} \, dx$$ is well-known, and can be looked up in books or on-line.

 

[shrub_broom]

First you need to prove that $\frac{sin(x)}{x}$ is trivial at $x = 0$, to calculate out integral$I = \int_0^{\infty} \frac {sin(x)} {x}$, you can first solve the integral $\int_0^{\infty} \frac{e^{-ax} sin(x)} {x} dx$, let $I(a) = \int_0^{\infty} \frac{e^{-ax} sin(x)} {x} dx$, then $I^{'}(a) = -\int_0^{\infty} e^{-ax} sin(x)dx = -\frac{1}{a^2+1}$ (1),which means $I(a) = -\arctan(a) + C$, since $I(\infty) = 0$, $C = \frac {\pi} {2}$. Hence, $\lim_{a \rightarrow 0} I(a) = I = \frac {\pi} {2}$(2). To know why (1) (2) are reasonable, you are advised to read Rudin's Principle. This integral can also be solved by Dirchlet Integral, which is more elementary and trivial.