Primitive of x by Logarithm of x squared plus a squared/Proof 1
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Theorem
- $\ds \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$
Proof
| \(\ds \int x \map \ln {x^2 + a^2} \rd x\) | \(=\) | \(\ds \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C\) | Primitive of $x^m \map \ln {x^2 + a^2}$ with $m = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {x^2 \map \ln {x^2 + a^2} } 2 - \paren {\frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} } + C\) | Primitive of $\dfrac {x^3} {x^2 + a^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C\) | simplifying |
$\blacksquare$