Question

How do you find `\int _ { 0} ^ { x } \frac { \sin ( t ) } { t } d t`?

Answer 1

Your integral cannot be expressed in terms of elementary functions.

Explanation:

Unfortunately, your integral cannot be computed using elementary functions.

A series solution can be computed in order to compute the integral to an arbitrary amount of accuracy, though.

`int_{0}^{x}(\sin(t))/(t) "d"t = \int_{0}^{x} (\sum_{n=1}^{\infty}(-1)^n(t^(2n+1))/((2n+1)!))/(t) "d"t`
`int_{0}^{x}(\sin(t))/(t) "d"t = int_{0}^{x} \sum_{n=1}^{\infty}(-1)^n(t^(2n))/((2n+1)!) "d"t`

`int_{0}^{x}(\sin(t))/(t) "d"t = \sum_{n=1}^{\infty}(-1)^n(x^(2n))/((2n+1)(2n+1)!)`.

Interestingly though, there is a closed form for the indefinite integral across the entire domain which I won't prove here but you can look up online.

`int_{-\infty}^{\infty}(sin(x))/x=\pi`