Primitive of Reciprocal of x squared plus a squared/Arctangent Form/Proof 3
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Theorem
- $\ds \int \frac {\d x} {x^2 + a^2} = \frac 1 a \arctan \frac x a + C$
Proof
| \(\ds \int \frac {\d x} {x^2 + a^2}\) | \(=\) | \(\ds \frac 1 a \int \frac {\d t} {t^2 + 1}\) | Substitution of $x \to a t$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} }\) | factoring | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \paren {\int \frac {\d t} {1 + i t} + \int \frac {\d t} {1 - i t} }\) | Definition of Partial Fractions Expansion | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \paren {i \map \ln {1 - i t} - i \map \ln {1 + i t} } + C\) | Primitive of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac i {2 a} \map \ln {\frac {1 - i t} {1 + i t} } + C\) | Sum of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \arctan \frac x a + C\) | Arctangent Logarithmic Formulation and substituting back $t \to \dfrac x a$ |
$\blacksquare$