Primitive of x squared over x squared plus a squared
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Theorem
- $\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$
where $a$ is a non-zero constant.
Proof
| \(\ds \int \frac {x^2 \rd x} {x^2 + a^2}\) | \(=\) | \(\ds \int \paren {1 - \frac {a^2} {x^2 + a^2} } \rd x\) | long division | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \d x - a^2 \int \frac {\d x} {x^2 + a^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds x - a^2 \int \frac {\d x} {x^2 + a^2} + C\) | Primitive of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds x - a^2 \frac 1 a \arctan {\frac x a} + C\) | Primitive of $\dfrac 1 {x^2 + a^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds x - a \arctan {\frac x a} + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 + a^2$: $14.127$
- 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: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.3.$