Primitive of Logarithm of x squared plus a squared

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Theorem

$\ds \int \map \ln {x^2 + a^2} \rd x = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C$


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \map \ln {x^2 + a^2}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {2 x} {x^2 + a^2}\) Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds x\) Primitive of Power


Then:

\(\ds \int \map \ln {x^2 + a^2} \rd x\) \(=\) \(\ds x \map \ln {x^2 + a^2} - \int \frac {2 x^2 \rd x} {x^2 + a^2} + C\) Integration by Parts
\(\ds \) \(=\) \(\ds x \map \ln {x^2 + a^2} - 2 \int \frac {x^2 \rd x} {x^2 + a^2} + C\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds x \map \ln {x^2 + a^2} - 2 \paren {x - a \arctan {\frac x a} } + C\) Primitive of $\dfrac {x^2} {x^2 + a^2}$
\(\ds \) \(=\) \(\ds x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C\) simplifying

$\blacksquare$


Also see


Sources