Primitive of x by Exponential of a x

Theorem

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

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

$\ds u$

$=$

$\ds x$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds 1$

and let:

$\ds \frac {\d v} {\d x}$

$=$

$\ds e^{a x}$

$\ds \leadsto \ \ $

$\ds v$

$=$

$\ds \frac {e^{a x} } a$

Then:

$\ds \int x e^{a x} \rd x$

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds x \paren {\frac {e^{a x} } a} - \frac 1 a \paren {\frac {e^{a x} } a} + C$

$\ds $

$=$

$\ds \frac {e^{a x} } a \paren {x - \frac 1 a} + C$

simplifying

$\blacksquare$

This result is also seen presented in the form:

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

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.510$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(25)$ Integrals Involving $e^{a x}$: $17.25.2.$