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

The integral of (e^{-5x}+3xe^x)/2 is 1/2 (-1/5 e^{-5x}+3(e^xx-e^x))+C

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integral of (e^{-5x}+3xe^x)/2

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

\frac{1}{2} (- \frac{1}{5} e^{- 5 x} + 3 (e^{x} x - e^{x})) + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int e^{- 5 x} d x + \int 3 x e^{x} d x)

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

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

= \frac{1}{2} (- \frac{1}{5} e^{- 5 x} + 3 (e^{x} x - e^{x}))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{5} e^{- 5 x} + 3 (e^{x} x - e^{x})) + C

x \to + 3 lim (\frac{1}{( x - 3 )})

y = 2 (12 x - 42)

2 t - \frac{8}{3}

z \to 0 lim (\frac{I m ( z )}{z})

\frac{2}{x}, 4

What is the integral of (e^{-5x}+3xe^x)/2 ?
The integral of (e^{-5x}+3xe^x)/2 is 1/2 (-1/5 e^{-5x}+3(e^xx-e^x))+C