Primitive of x cubed over x squared plus a squared squared/Proof 2
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Theorem
- $\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$
Proof
| \(\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2}\) | \(=\) | \(\ds \int \frac {x \paren {x^2 + a^2 - a^2} } {\paren {x^2 + a^2}^2} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {x \paren {x^2 + a^2} } {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {x \rd x} {x^2 + a^2} - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2}\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \map \ln {x^2 + a^2} - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2} + C\) | Primitive of $\dfrac x {x^2 + a^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \map \ln {x^2 + a^2} - a^2 \paren {\frac {-1} {2 \paren {x^2 - a^2} } } + C\) | Primitive of $\dfrac x {\paren {x^2 + a^2}^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C\) | simplifying |
$\blacksquare$