Primitive of x over x squared plus a squared/Proof 1

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

$\blacksquare$