Derivative of improper Integral $f(t)= \int_0^1 \frac{\sin(xt)}{x}\:dx$

Question

I'm looking for the derivative of the following function $$f(t)= \int_0^1 \frac{\sin(xt)}{x}~dx$$

I tried to use $\displaystyle \frac{1}{x} =\int_0^{\infty}e^{-xy}dy$ and then by using Fubini changed the order of the integrals but that didn't solve the problem.

Answer 1

Hint. The integrand may be extended as a differentiable function, one may then use the Leibniz integral formula giving $$ f'(t)= \int_0^1 x \cdot \frac{\cos(xt)}{x}~dx=\int_0^1 \cos(xt)~dx=\frac{\sin t}{t},\qquad t>0. $$