# inte^x * lnx dx?

Jan 20, 2018

$\textcolor{b l u e}{{e}^{x} \cdot \ln \left(x\right) - E i \left(x\right) + C}$, where

color(blue)(Ei(x) represents the exponential integral

$\textcolor{g r e e n}{{\int}_{- \infty}^{x} {x}^{t} / t \mathrm{dt}}$

#### Explanation:

Given:

$\textcolor{b r o w n}{\int \text{ "e^xln(x) " } \mathrm{dx}}$ ... Expression.1

Integration by Parts: $\textcolor{g r e e n}{\int \text{ " f*g' = fg - int" } f ' g}$

We will integrate by parts

Referring to our problem, we have

color(brown)(f=ln(x) and g = e^x" "

While solving our problem,

we will consider the following known results in Calculus:

$\textcolor{b r o w n}{f ' = \frac{1}{x} \mathmr{and} g ' = {e}^{x}}$ and

$\textcolor{b r o w n}{\int \text{ } {e}^{x} \mathrm{dx} = {e}^{x} + C}$

Now, we can write our Expression.1 as

$\ln \left(x\right) \cdot {e}^{x} - \int \text{ "[1/x*e^x]" } \mathrm{dx}$

$\Rightarrow {e}^{x} \cdot \ln \left(x\right) - \int \text{ } {e}^{x} / x \cdot \mathrm{dx}$ ... Expression.2

$\textcolor{b r o w n}{- - - - - - - - - - - - - - - - - - - -}$

Note:

color(blue)(Ei(x) represents the exponential integral

$\textcolor{g r e e n}{{\int}_{- \infty}^{x} {x}^{t} / t \mathrm{dt}}$

For color(red)(x>0, the integral color(red)(Ei(x) is interpreted as Cauchy Principal Value

$\textcolor{b r o w n}{- - - - - - - - - - - - - - - - - - - -}$

We will use the above note on $\textcolor{red}{E i \left(x\right)}$ in writing our final solution

We rewrite our ... Expression.2 as

$\textcolor{b l u e}{{e}^{x} \cdot \ln \left(x\right) - E i \left(x\right) + C}$

Hope you find this solution useful.