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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.166$