Primitive of x squared over a squared minus x squared/Logarithm Form

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$

for $x^2 < a^2$.


Proof

\(\displaystyle \int \frac {x^2 \rd x} {a^2 - x^2}\) \(=\) \(\displaystyle \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {- \left({a^2 - x^2}\right)} {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle - \int \d x + a^2 \int \frac {\d x} {a^2 - x^2}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle -x + a^2 \int \frac {\d x} {a^2 - x^2} + C\) Primitive of Constant
\(\displaystyle \) \(=\) \(\displaystyle -x + a^2 \paren {\frac 1 {2 a} \map \ln {\frac {a + x} {a - x} } } + C\) Primitive of $\dfrac 1 {a^2 - x^2}$: Logarithm Form
\(\displaystyle \) \(=\) \(\displaystyle -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C\) simplifying

$\blacksquare$


Sources