Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1
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Theorem
Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$.
Then:
- $\ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\ & \\ \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$
Proof
Case where $\size x < a$
Let $\size x < a$.
Then:
- $\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C$
Let $\size x < a$.
Then:
\(\ds \int \frac {\d x} {x^2 - a^2}\) | \(=\) | \(\ds -\frac 1 a \artanh {\frac x a} + C\) | Primitive of $\dfrac 1 {x^2 - a^2}$: $\artanh$ form | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C\) | $\artanh \dfrac x a$ in Logarithm Form | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C\) | Logarithm of Reciprocal |
Case where $\size x > a$
Let $\size x > a$.
Then:
- $\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C$
Let $\size x > a$.
Then:
\(\ds \int \frac {\d x} {x^2 - a^2}\) | \(=\) | \(\ds -\frac 1 a \arcoth \frac x a + C\) | Primitive of $\dfrac 1 {x^2 - a^2}$: $\arcoth$ form | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C\) | $\arcoth \dfrac x a$ in Logarithm Form | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 {2 a} \map \ln {\frac {x + a} {x - a} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2 a} \map \ln {\frac {x - a} {x + a} } + C\) | Reciprocal of Logarithm |
$\blacksquare$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals