Solving Integral Equations: Find x from 1-x+ ∫^x_1 (sin t/t) dt

[Rectifier]

The problem
I want to find $x$ which solves $1-x+ \int^x_1 \frac{\sin t}{t} \ dt = 0$

The attempt
$\int^x_1 \frac{\sin t}{t} \ dt = x -1$ I see that the answer is $x=1$ but I want to be able to calculate it mechanically in case if I get similar problem with other elements. Any suggestions on how I can do that?

 

[BvU]

Hi,
'mechanically' sounds good. But, ${\sin x\over x}$ is (and I https://owlcation.com/stem/How-to-Integrate-sinxx-and-cosxx )

one of the simplest examples of non-integrable functions in the sense that their antiderivatives cannot be expressed in terms of elementary functions, in other words, they don't have closed-form antiderivatives.​

However, apart from $x=1$ there shouldn't be too many other solutions ... $x-1$ grows faster than the integral.
You could also investigate domain [0,x] : with ${\sin x\over x} < 1$ the integral is always different from x-1.

 

[Rectifier]

Okay, thank your for your help.