Indefinite Integral

**th%$&873** · December 10th 2007, 07:13 AM

What is ∫(sin(√(x))/x)dx?

**curvature** · December 10th 2007, 08:19 AM

Originally Posted by th%$&873

What is ∫(sin(√(x))/x)dx?

This integral is unsolvable (no finite forms).

**topsquark** · December 10th 2007, 08:30 AM

Originally Posted by th%$&873

What is ∫(sin(√(x))/x)dx?

The best we can do is to let $u = \sqrt{x} \implies 2u~du = dx$ , so
$\int \frac{sin(\sqrt{x})}{x}~dx = \int \frac{sin(u)}{u^2} \cdot 2u~du = 2 \int \frac{sin(u)}{u}~du$

This integral (at least) sometimes goes by the name of "Si." (So as a function of x it is $Si(x)$ .) There is no closed-form solution to this integral.

-Dan

**Krizalid** · December 10th 2007, 10:19 AM

Originally Posted by topsquark

This integral (at least) sometimes goes by the name of "Si."

The $\text{Si}$ function is defined as follows:

${\text{Si}}\,(x) = \int_0^x {\frac{{\sin u}}<br /> {u}\,du} .$

As a definite integral, may be it could have more sense.

**th%$&873** · December 13th 2007, 12:00 PM

Thanks, everyone.

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