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

The integral of x^2e^{kx} is e^{kx} 1/k x^2-2/k (e^{kx} 1/k x-e^{kx} 1/(k^2))+C

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integral of x^2e^{kx}

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

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

Solution steps

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

Apply Integration By Parts

= e^{k x} \frac{1}{k} x^{2} - \int e^{k x} \frac{2}{k} x d x

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

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

Add a constant to the solution

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

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