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

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

Popular >

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

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

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

Solution steps

\int ln (x) e^{- 2 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

f (x) = \frac{2 x - 1}{x + 5}, at x = 3

\frac{d y}{d x} - y = 1, y (0) = 1

f (x) = \frac{1}{x ^{2}}, at x = 3

\int \frac{5 x ^{3} + 5 x + 11}{( x ^{2} + 1 ) ^{2}} d x

y^{^{'}} - \frac{y}{x} = x^{2}

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