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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

\int - \frac{2}{3} e^{- 3 x} x d x = - \frac{2}{27} (- 3 e^{- 3 x} x - e^{- 3 x})

= 3 (- \frac{1}{3} e^{- 3 x} x^{2} - (- \frac{2}{27} (- 3 e^{- 3 x} x - e^{- 3 x})))

Simplify

3 (- \frac{1}{3} e^{- 3 x} x^{2} - (- \frac{2}{27} (- 3 e^{- 3 x} x - e^{- 3 x}))) : - e^{- 3 x} x^{2} - \frac{2}{3} e^{- 3 x} x - \frac{2}{9} e^{- 3 x}

= - e^{- 3 x} x^{2} - \frac{2}{3} e^{- 3 x} x - \frac{2}{9} e^{- 3 x}

Add a constant to the solution

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

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