What is the integral of e^{-2x}ln(x) ?

The integral of e^{-2x}ln(x) is -1/2 e^{-2x}ln(x)+1/2 \Ei(-2x)+C

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integral of e^{-2x}ln(x)

\int e^{- 2 x} ln (x) d x

- \frac{1}{2} e^{- 2 x} ln (x) + \frac{1}{2} Ei (- 2 x) + C

Solution steps

\int e^{- 2 x} ln (x) d x

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} ln (x) - \int - \frac{e ^{- 2 x}}{2 x} d x

= - \frac{1}{2} e^{- 2 x} ln (x) - \int - \frac{e ^{- 2 x}}{2 x} d x

\int - \frac{e ^{- 2 x}}{2 x} d x = - \frac{1}{2} Ei (- 2 x)

= - \frac{1}{2} e^{- 2 x} ln (x) - (- \frac{1}{2} Ei (- 2 x))

Simplify

= - \frac{1}{2} e^{- 2 x} ln (x) + \frac{1}{2} Ei (- 2 x)

Add a constant to the solution

= - \frac{1}{2} e^{- 2 x} ln (x) + \frac{1}{2} Ei (- 2 x) + C

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What is the integral of e^{-2x}ln(x) ?
The integral of e^{-2x}ln(x) is -1/2 e^{-2x}ln(x)+1/2 \Ei(-2x)+C