Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 2
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Theorem
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
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 |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration: $\text {I}$.