Primitive of x by Exponential of a x/Also presented as

Primitive of $x e^{a x} \rd x$: Also presented as

This result is also seen presented in the form:

$\ds \int x e^{a x} \rd x = \frac {e^{a x} } {a^2} \paren {a x - 1} + C$

Proof

\(\ds \frac {e^{a x} } {a^2} \paren {a x - 1}\)

\(=\)

\(\ds \frac {e^{a x} } a \cdot \dfrac 1 a \paren {a x - 1}\)

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } a \paren {x - \dfrac 1 a}\)

$\blacksquare$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $104$.