Question

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

Answer 1

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

Explanation:

`int_4^(x^2)sin(e^t)dt`
Let `y=e^t`
`t=lny`
`dt=(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`
\0/ Here's our answer!