# How do you integrate f(x)=intsin(e^t)dt between 4 to x^2?

May 22, 2018

int_4^(x^2)sin(e^t)dt≈int _0^(+oo)sin(y)/ydy=pi/2

#### Explanation:

${\int}_{4}^{{x}^{2}} \sin \left({e}^{t}\right) \mathrm{dt}$
Let $y = {e}^{t}$
$t = \ln y$
$\mathrm{dt} = \frac{\mathrm{dy}}{y}$
int_4^(x^2)sin(e^t)dt=int_(e^4)^(e^(x²))sin(y)/ydy, and we can see that there's no result of this integration. However, using Dirichlet integral:
(int_(e^4)^(e^(x²))sin(y)/ydy) _(x to oo)≈int _0^(+oo)sin(y)/ydy=pi/2