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

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

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integral of x*e^{ax}

\int x \cdot e^{a x} d x

e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}} + C

Solution steps

\int x e^{a x} d x

Apply Integration By Parts

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

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

= e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}}

Add a constant to the solution

= e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}} + C

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