Primitive of Reciprocal of x by a squared minus x squared
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Theorem
- $\ds \int \frac {\d x} {x \paren {a^2 - x^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$.
Proof
| \(\ds \int \frac {\d x} {x \paren {a^2 - x^2} }\) | \(=\) | \(\ds \int \paren {\frac 1 {a^2 x} + \frac x {a^2 \paren {a^2 - x^2} } } \rd x\) | Partial Fraction Expansion | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} x + \frac 1 {a^2} \int \frac {x \rd x} {a^2 - x^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \ln \size x + \frac 1 {a^2} \int \frac {x \rd x} {a^2 - x^2} + C\) | Primitive of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \ln \size x + \frac 1 {a^2} \paren {-\frac 1 2 \map \ln {a^2 - x^2} } + C\) | Primitive of $\dfrac x {a^2 - x^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 a^2} \ln \size {x^2} - \frac 1 {2 a^2} \map \ln {a^2 - x^2} + C\) | Logarithm of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 a^2} \map \ln {x^2} - \frac 1 {2 a^2} \map \ln {a^2 - x^2} + C\) | Square of Real Number is Non-Negative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 - a^2} } + C\) | Difference of Logarithms |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.167$