What is the integral of 5x^2e^{3x} ?

The integral of 5x^2e^{3x} is 5(1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x}))+C

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integral of 5x^2e^{3x}

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

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

Solution steps

\int 5 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

= 5 \cdot \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

= 5 (\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})

= 5 (\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

= 5 (\frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x})) + C

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

\int 1 - x^{2} \cdot x d x

\int e^{x^{2}} - 1 d x

\int x - 2 x^{2} d x

\int (- \frac{2}{x}) d x

What is the integral of 5x^2e^{3x} ?
The integral of 5x^2e^{3x} is 5(1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x}))+C