inte^x * lnx dx?

1 Answer
Jan 20, 2018

color(blue)(e^x*ln(x)-Ei(x)+C), where

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

color(green)(int_(-oo)^x x^t/t dt)

Explanation:

Given:

color(brown)(int " "e^xln(x) " "dx) ... Expression.1

Integration by Parts: color(green)(int " " 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:

color(brown)(f' = 1/x and g' = e^x) and

color(brown)(int " "e^x dx = e^x + C)

Now, we can write our Expression.1 as

ln(x)*e^x - int " "[1/x*e^x]" " dx

rArr e^x*ln(x) - int " "e^x/x * dx ... Expression.2

color(brown)(--------------------)

Note:

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

color(green)(int_(-oo)^x x^t/t dt)

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

color(brown)(--------------------)

We will use the above note on color(red)(Ei(x)) in writing our final solution

We rewrite our ... Expression.2 as

color(blue)(e^x*ln(x)-Ei(x)+C)

Hope you find this solution useful.