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

The integral of 5x^2e^{5x} is e^{5x}x^2-2/5 e^{5x}x+2/25 e^{5x}+C

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

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

e^{5 x} x^{2} - \frac{2}{5} e^{5 x} x + \frac{2}{25} e^{5 x} + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{5} e^{5 x} x^{2} - \int \frac{2}{5} e^{5 x} x d x

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

\int \frac{2}{5} e^{5 x} x d x = \frac{2}{125} (5 e^{5 x} x - e^{5 x})

= 5 (\frac{1}{5} e^{5 x} x^{2} - \frac{2}{125} (5 e^{5 x} x - e^{5 x}))

Simplify

5 (\frac{1}{5} e^{5 x} x^{2} - \frac{2}{125} (5 e^{5 x} x - e^{5 x})) : e^{5 x} x^{2} - \frac{2}{5} e^{5 x} x + \frac{2}{25} e^{5 x}

= e^{5 x} x^{2} - \frac{2}{5} e^{5 x} x + \frac{2}{25} e^{5 x}

Add a constant to the solution

= e^{5 x} x^{2} - \frac{2}{5} e^{5 x} x + \frac{2}{25} e^{5 x} + C

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What is the integral of 5x^2e^{5x} ?
The integral of 5x^2e^{5x} is e^{5x}x^2-2/5 e^{5x}x+2/25 e^{5x}+C