What is the integral of x/2 e^{-(3x)/4} ?

The integral of x/2 e^{-(3x)/4} is -2/9 e^{-(3x)/4}(3x+4)+C

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integral of x/2 e^{-(3x)/4}

\int \frac{x}{2} e^{- \frac{3 x}{4}} d x

- \frac{2}{9} e^{- \frac{3 x}{4}} (3 x + 4) + C

Solution steps

\int \frac{x}{2} e^{- \frac{3 x}{4}} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x e^{- \frac{3 x}{4}} d x

Apply Integration By Parts

: - \frac{4}{3} e^{- \frac{3 x}{4}} x - \int - \frac{4}{3} e^{- \frac{3 x}{4}} d x

= \frac{1}{2} (- \frac{4}{3} e^{- \frac{3 x}{4}} x - \int - \frac{4}{3} e^{- \frac{3 x}{4}} d x)

\int - \frac{4}{3} e^{- \frac{3 x}{4}} d x = \frac{16}{9} e^{- \frac{3 x}{4}}

= \frac{1}{2} (- \frac{4}{3} e^{- \frac{3 x}{4}} x - \frac{16}{9} e^{- \frac{3 x}{4}})

Simplify

\frac{1}{2} (- \frac{4}{3} e^{- \frac{3 x}{4}} x - \frac{16}{9} e^{- \frac{3 x}{4}}) : - \frac{2}{9} e^{- \frac{3 x}{4}} (3 x + 4)

= - \frac{2}{9} e^{- \frac{3 x}{4}} (3 x + 4)

Add a constant to the solution

= - \frac{2}{9} e^{- \frac{3 x}{4}} (3 x + 4) + C

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What is the integral of x/2 e^{-(3x)/4} ?
The integral of x/2 e^{-(3x)/4} is -2/9 e^{-(3x)/4}(3x+4)+C