1. integral for cos(xt)/(1+t) dt

cost(xt)/1+t dt anyone can help

2. i need to show it with leibnez rule but i cant find how

3. Lets start evaluating the indefinite integral...

$\varphi (x, t)= \int \frac{e^{x t}}{1+t}\cdot dt$ (1)

Setting $1+t=\tau$ You obtain first the integral...

$\varphi (x,\tau) = e^{-x} \int \frac{e^{x \tau}}{\tau}\cdot d\tau$ (2)

... and now setting $x \tau = \xi$...

$\varphi (x,\xi)= e^{-x} \int \frac{e^{\xi}}{\xi}\cdot d\xi = e^{-x}\cdot Ei(\xi) + c$ (3)

... where $Ei(*)$ is the socalled 'Integral exponential function'. What we have obtained is that...

$\int \frac{e^{x t}}{1+t}\cdot dt = e^{-x}\cdot Ei [x\cdot (1+t)] + c$ (4)

If now we write in (4) $i x$ instead of $x$ we obtain...

$\int \frac{\cos x t}{1+t}\cdot dt = Re \{ e^{-i x}\cdot Ei[i x \cdot (1+t)]\} + c$ (5)

Kind regards

$\chi$ $\sigma$