What is the integral of xe^{-2/5 x} ?

The integral of xe^{-2/5 x} is -5/2 e^{-(2x)/5}x-25/4 e^{-(2x)/5}+C

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integral of xe^{-2/5 x}

\int x e^{- \frac{2}{5} x} d x

- \frac{5}{2} e^{- \frac{2 x}{5}} x - \frac{25}{4} e^{- \frac{2 x}{5}} + C

Solution steps

\int x e^{- \frac{2}{5} x} d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \int x e^{- \frac{2 x}{5}} d x

Apply Integration By Parts

: \frac{1}{2} e^{- \frac{2 x}{5}} x^{2} - \int - \frac{1}{5} e^{- \frac{2 x}{5}} x^{2} d x

= \frac{1}{2} e^{- \frac{2 x}{5}} x^{2} - \int - \frac{1}{5} e^{- \frac{2 x}{5}} x^{2} d x

\int - \frac{1}{5} e^{- \frac{2 x}{5}} x^{2} d x = \frac{1}{4} (2 e^{- \frac{2 x}{5}} x^{2} - 25 (- \frac{2 e ^{- \frac{2 x}{5}} x}{5} - e^{- \frac{2 x}{5}}))

= \frac{1}{2} e^{- \frac{2 x}{5}} x^{2} - \frac{1}{4} (2 e^{- \frac{2 x}{5}} x^{2} - 25 (- \frac{2 e ^{- \frac{2 x}{5}} x}{5} - e^{- \frac{2 x}{5}}))

Simplify

\frac{1}{2} e^{- \frac{2 x}{5}} x^{2} - \frac{1}{4} (2 e^{- \frac{2 x}{5}} x^{2} - 25 (- \frac{2 e ^{- \frac{2 x}{5}} x}{5} - e^{- \frac{2 x}{5}})) : - \frac{5}{2} e^{- \frac{2 x}{5}} x - \frac{25}{4} e^{- \frac{2 x}{5}}

= - \frac{5}{2} e^{- \frac{2 x}{5}} x - \frac{25}{4} e^{- \frac{2 x}{5}}

Add a constant to the solution

= - \frac{5}{2} e^{- \frac{2 x}{5}} x - \frac{25}{4} e^{- \frac{2 x}{5}} + C

\int \frac{1}{( x - 1 ) ( x ^{2} + 1 )} d x

y^{^{''}} - 5 y^{^{'}} - 6 y = 12 e^{2} t

\frac{\partial}{\partial x} (y e^{3 x})

\frac{d}{d x} (- 2 (2 x^{2} - 4 x - 1)^{2})

What is the integral of xe^{-2/5 x} ?
The integral of xe^{-2/5 x} is -5/2 e^{-(2x)/5}x-25/4 e^{-(2x)/5}+C