What is the integral of x^2*e^{3x} ?

The integral of x^2*e^{3x} is 1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x})+C

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integral of x^2*e^{3x}

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

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

Solution steps

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

Apply Integration By Parts

: \frac{1}{3} e^{3 x} x^{2} - \int \frac{2}{3} e^{3 x} x d x

= \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})

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

Add a constant to the solution

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

\frac{d}{d x} (4 3 x^{2} - 5 x + 2)

y^{^{''}} + 2 y^{^{'}} + y = 4 e^{- t}, y (0) = 2, y^{^{'}} (0) = - 1

x^{^{'}} = x^{2} + 1

y^{^{''}} + 6 y^{^{'}} - 5 y = 0

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

What is the integral of x^2*e^{3x} ?
The integral of x^2*e^{3x} is 1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x})+C