Primitive of x squared by Exponential of a x
Jump to navigation
Jump to search
Theorem
- $\ds \int x^2 e^{a x} \rd x = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C$
Proof
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:
\(\ds u\) | \(=\) | \(\ds x^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds 2 x\) | Derivative of Power |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds e^{a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {e^{a x} } a\) | Primitive of $e^{a x}$ |
Then:
\(\ds \int x^2 e^{a x} \rd x\) | \(=\) | \(\ds x^2 \paren {\frac {e^{a x} } a} - \int 2 x \frac {e^{a x} } a \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds x^2 \paren {\frac {e^{a x} } a} - \frac 2 a \int x e^{a x} \rd x + C\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds x^2 \paren {\frac {e^{a x} } a} - \frac 2 a \paren {\frac {e^{a x} } a \paren {x - \frac 1 a} } + C\) | Primitive of $x e^{a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C\) | simplifying |
$\blacksquare$
Examples
Primitive of $x^2 e^{-x}$
- $\ds \int x^2 e^{-x} \rd x = -e^{-x} \paren {x^2 + 2 x + 2} + C$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.511$