Primitive of x squared over a squared minus x squared/Logarithm Form
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Theorem
- $\ds \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$.
Proof
| \(\ds \int \frac {x^2 \rd x} {a^2 - x^2}\) | \(=\) | \(\ds \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {-\paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\int \d x + a^2 \int \frac {\d x} {a^2 - x^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds -x + a^2 \int \frac {\d x} {a^2 - x^2} + C\) | Primitive of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds -x + a^2 \paren {\frac 1 {2 a} \map \ln {\frac {a + x} {a - x} } } + C\) | Primitive of $\dfrac 1 {a^2 - x^2}$: Logarithm Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.165$