What is the integral of xe^{ax} ?

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

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integral of xe^{ax}

\int x 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

= 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

2 y^{^{''}} - 3 y^{^{'}} + y = 0, y (0) = 2, y^{^{'}} (0) = \frac{1}{2}

\int t^{2} t + 8 d t

y = \frac{2 x}{x - 1}

x \to 0 lim ((\frac{2 arccos ( x )}{π})^{\frac{1}{x}})

\int \frac{2 + x + x ^{2}}{x} d x

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