Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 3

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:

\(\ds \int \frac {\d x} {a^2 - x^2}\)

\(=\)

\(\ds -\int \frac {\d x} {x^2 - a^2}\)

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

$\blacksquare$