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