Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form
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Theorem
- $\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$
for $x^2 > a^2$.
Proof
Let:
\(\ds \int \frac {x^2 \rd x} {x^2 - a^2}\) | \(=\) | \(\ds \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds x + a^2 \int \frac {\d x} {x^2 - a^2} + C\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds x + a^2 \paren {-\frac 1 a \coth^{-1} {\frac x a} } + C\) | Primitive of $\dfrac 1 {x^2 - a^2}$: $\coth^{-1}$ form | |||||||||||
\(\ds \) | \(=\) | \(\ds x - a \coth^{-1} {\frac x a} + C\) | simplifying |
$\blacksquare$