Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a
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Theorem
Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $\size x < a$.
Then:
- $\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$
Proof 1
Let $\size x < a$.
Then:
\(\ds \int \frac {\d x} {a^2 - x^2}\) | \(=\) | \(\ds \frac 1 a \artanh {\frac x a} + C\) | Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ 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 |
Proof 2
Let $\size x < a$.
Then:
\(\ds \int \frac {\d x} {a^2 - x^2}\) | \(=\) | \(\ds \int \frac {\d x} {\paren {a + x} \paren {a - x} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \int \frac {\d x} {a + x} + \frac 1 {2 a} \int \frac {\d x} {a - x}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \ln \size {a + x} - \frac 1 {2 a} \ln \size {a - x} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \map \ln {a + x} - \frac 1 {2 a} \map \ln {a - x} + C\) | as both $a + x < 0$ and $a - x < 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C\) | Difference of Logarithms |
$\blacksquare$
Proof 3
Let $\size x < a$.
Then:
\(\ds \int \frac {\d x} {a^2 - x^2}\) | \(=\) | \(\ds -\int \frac {\d x} {x^2 - a^2}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C\) | Primitive of $\dfrac 1 {x^2 - a^2}$ for $\size x < a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \map \ln {\frac {x + a} {x - a} } + C\) | Logarithm of Reciprocal |
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $16$.
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(vii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.41$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.163$