What is the integral of e^{-4x}ln(3x) ?

The integral of e^{-4x}ln(3x) is -1/4 e^{-4x}ln(3x)+1/4 \Ei(-4x)+C

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integral of e^{-4x}ln(3x)

\int e^{- 4 x} ln (3 x) d x

- \frac{1}{4} e^{- 4 x} ln (3 x) + \frac{1}{4} Ei (- 4 x) + C

Solution steps

\int e^{- 4 x} ln (3 x) d x

Apply Integration By Parts

= - \frac{1}{4} e^{- 4 x} ln (3 x) - \int - \frac{e ^{- 4 x}}{4 x} d x

\int - \frac{e ^{- 4 x}}{4 x} d x = - \frac{1}{4} Ei (- 4 x)

= - \frac{1}{4} e^{- 4 x} ln (3 x) - (- \frac{1}{4} Ei (- 4 x))

Simplify

= - \frac{1}{4} e^{- 4 x} ln (3 x) + \frac{1}{4} Ei (- 4 x)

Add a constant to the solution

= - \frac{1}{4} e^{- 4 x} ln (3 x) + \frac{1}{4} Ei (- 4 x) + C

What is the integral of e^{-4x}ln(3x) ?
The integral of e^{-4x}ln(3x) is -1/4 e^{-4x}ln(3x)+1/4 \Ei(-4x)+C