Primitive of x over x squared plus a squared/Proof 1
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Theorem
- $\ds \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$
Proof
\(\ds u\) | \(=\) | \(\ds x^2 + a^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds 2 x\) | Power Rule for Derivatives and Derivative of Constant | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {x \rd x} {x^2 + a^2}\) | \(=\) | \(\ds \frac 1 2 \ln \size {x^2 + a^2} + C\) | Primitive of Function under its Derivative | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \, \map \ln {x^2 + a^2} + C\) | Absolute Value of Even Power‎ |
$\blacksquare$