Primitive of x cubed over a squared minus x squared

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Theorem

$\ds \int \frac {x^3 \rd x} {a^2 - x^2} = -\frac {x^2} 2 - \frac {a^2} 2 \map \ln {a^2 - x^2} + C$

for $x^2 < a^2$.


Proof

Let:

\(\ds \int \frac {x^3 \rd x} {a^2 - x^2}\) \(=\) \(\ds \int \frac {x \paren {x^2 - a^2 + a^2} } {a^2 - x^2} \rd x\)
\(\ds \) \(=\) \(\ds \int \frac {-x \paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2 x} {a^2 - x^2} \rd x\)
\(\ds \) \(=\) \(\ds -\int x \rd x + a^2 \int \frac {x \rd x} {a^2 - x^2}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds -\frac {x^2} 2 + a^2 \int \frac {x \rd x} {a^2 - x^2} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds -\frac {x^2} 2 + a^2 \paren {-\frac 1 2 \map \ln {a^2 - x^2} } + C\) Primitive of Reciprocal of $\dfrac x {a^2 - x^2}$
\(\ds \) \(=\) \(\ds -\frac {x^2} 2 - \frac {a^2} 2 \map \ln {a^2 - x^2} + C\) simplifying

$\blacksquare$


Also see


Sources