Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary
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Theorem
- $\ds \int \frac {\d x} {-\sqrt {x^2 - a^2} } = \ln \size {x - \sqrt {x^2 - a^2} } + C$
for $\size x > a$.
Proof
| \(\ds \int \frac {\d x} {\sqrt {x^2 - a^2} }\) | \(=\) | \(\ds \ln \size {x + \sqrt {x^2 - a^2} } + C\) | Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ in Logarithm Form | |||||||||||
| \(\ds \int \frac {\d x} {-\sqrt {x^2 + a^2} }\) | \(=\) | \(\ds -\ln \size {x + \sqrt {x^2 - a^2} } + C\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {x - \sqrt {x^2 - a^2} } + C + \map \ln {a^2}\) | Negative of Logarithm of x plus Root x squared minus a squared | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {x - \sqrt {x^2 - a^2} } + C\) | subsuming $\map \ln {a^2}$ into the constant |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form
- Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $15$.