What is the integral of 2/3 xe^{-3/5 x} ?

The integral of 2/3 xe^{-3/5 x} is 2/3 (-5/3 e^{-(3x)/5}x-25/9 e^{-(3x)/5})+C

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integral of 2/3 xe^{-3/5 x}

\int \frac{2}{3} x e^{- \frac{3}{5} x} d x

\frac{2}{3} (- \frac{5}{3} e^{- \frac{3 x}{5}} x - \frac{25}{9} e^{- \frac{3 x}{5}}) + C

Solution steps

\int \frac{2}{3} x e^{- \frac{3}{5} x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{3} \cdot \int x e^{- \frac{3}{5} x} d x

Multiply fractions:

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

= \frac{2}{3} \cdot \int x e^{- \frac{3 x}{5}} d x

Apply Integration By Parts

: - \frac{5}{3} e^{- \frac{3 x}{5}} x - \int - \frac{5}{3} e^{- \frac{3 x}{5}} d x

= \frac{2}{3} (- \frac{5}{3} e^{- \frac{3 x}{5}} x - \int - \frac{5}{3} e^{- \frac{3 x}{5}} d x)

\int - \frac{5}{3} e^{- \frac{3 x}{5}} d x = \frac{25}{9} e^{- \frac{3 x}{5}}

= \frac{2}{3} (- \frac{5}{3} e^{- \frac{3 x}{5}} x - \frac{25}{9} e^{- \frac{3 x}{5}})

Add a constant to the solution

= \frac{2}{3} (- \frac{5}{3} e^{- \frac{3 x}{5}} x - \frac{25}{9} e^{- \frac{3 x}{5}}) + C

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What is the integral of 2/3 xe^{-3/5 x} ?
The integral of 2/3 xe^{-3/5 x} is 2/3 (-5/3 e^{-(3x)/5}x-25/9 e^{-(3x)/5})+C