Primitive of Reciprocal of x by a squared minus x squared

From ProofWiki
Jump to navigation Jump to search
$\displaystyle \int \frac {\mathrm d x} {x \left({a^2 - x^2}\right)} = \frac 1 {2 a^2} \ln \left({\frac {x^2} {a^2 - x^2} }\right) + C$

for $x^2 < a^2$.


Proof

\(\displaystyle \int \frac {\mathrm d x} {x \left({a^2 - x^2}\right)}\) \(=\) \(\displaystyle \int \left({\frac 1 {a^2 x} + \frac x {a^2 \left({a^2 - x^2}\right)} }\right) \ \mathrm d x\) Partial Fraction Expansion
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \int \frac {\mathrm d x} x + \frac 1 {a^2} \int \frac {x \ \mathrm d x} {a^2 - x^2}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \ln \left\vert{x}\right\vert + \frac 1 {a^2} \int \frac {x \ \mathrm d x} {a^2 - x^2} + C\) Primitive of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \ln \left\vert{x}\right\vert + \frac 1 {a^2} \left({-\frac 1 2 \ln \left ({a^2 - x^2}\right) }\right) + C\) Primitive of $\dfrac x {a^2 - x^2}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a^2} \ln \left\vert{x^2}\right\vert - \frac 1 {2 a^2} \ln \left ({a^2 - x^2}\right) + C\) Logarithm of Power
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a^2} \ln \left({x^2}\right) - \frac 1 {2 a^2} \ln \left ({a^2 - x^2}\right) + C\) as $x^2 > 0$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a^2} \ln \left({\frac {x^2} {x^2 - a^2} }\right) + C\) Difference of Logarithms

$\blacksquare$


Also see


Sources