Primitive of Logarithm of x squared minus a squared
Jump to navigation
Jump to search
Theorem
- $\ds \int \map \ln {x^2 - a^2} \rd x = x \map \ln {x^2 - a^2} - 2 x + a \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$.
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:
\(\ds u\) | \(=\) | \(\ds \map \ln {x^2 - a^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {2 x} {x^2 - a^2}\) | Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds x\) | Primitive of Power |
Then:
\(\ds \int \map \ln {x^2 - a^2} \rd x\) | \(=\) | \(\ds x \map \ln {x^2 - a^2} - \int \frac {2 x^2 \rd x} {x^2 - a^2} + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds x \map \ln {x^2 - a^2} - 2 \int \frac {x^2 \rd x} {x^2 - a^2} + C\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds x \map \ln {x^2 - a^2} - 2 \paren {x + \frac a 2 \map \ln {\frac {x - a} {x + a} } } + C\) | Primitive of $\dfrac {x^2} {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \map \ln {x^2 - a^2} - 2 x - a \map \ln {\frac {x + a} {x - a} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds x \map \ln {x^2 - a^2} - 2 x + a \map \ln {\frac {x + a} {x - a} } + C\) | Logarithm of Reciprocal |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\ln x$: $14.538$