Primitive of x cubed over x squared minus a squared
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Theorem
- $\displaystyle \int \frac {x^3 \ \mathrm d x} {x^2 - a^2} = \frac {x^2} 2 + \frac {a^2} 2 \ln \left({x^2 - a^2}\right) + C$
for $x^2 > a^2$.
Proof
Let:
\(\ds \int \frac {x^3 \ \mathrm d x} {x^2 - a^2}\) | \(=\) | \(\ds \int \frac {x \left({x^2 - a^2 + a^2}\right)} {x^2 - a^2} \ \mathrm d x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {x \left({x^2 - a^2}\right)} {x^2 - a^2} \ \mathrm d x + \int \frac {a^2 x} {x^2 - a^2} \ \mathrm d x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int x \ \mathrm d x + a^2 \int \frac {x \ \mathrm d x} {x^2 - a^2}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 + a^2 \int \frac {x \ \mathrm d x} {x^2 - a^2} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 + a^2 \left({\frac 1 2 \ln \left({x^2 - a^2}\right)}\right) + C\) | Primitive of $\dfrac x {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 + \frac {a^2} 2 \ln \left({x^2 - a^2}\right) + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.147$