Primitive of Reciprocal of a x squared plus b x plus c/Examples

Examples of Use of Primitive of $\dfrac 1 {a x^2 + b x + c}$

Primitive of $\dfrac 1 {3 x^2 + 4 x + 2}$

$\ds \int \frac {\d x} {3 x^2 + 4 x + 2} = \dfrac 1 {\sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C$

Primitive of $\dfrac 1 {x^2 + 4 x + 5}$

$\ds \int \dfrac {\d x} {x^2 + 4 x + 5} = \map \arctan {x + 2} + C$

Primitive of $\dfrac 1 {x^2 + 2 a x + b}$

$\ds \int \frac {\d x} {x^2 + 2 a x + b} = \dfrac 1 {\sqrt {b - a^2} } \map \arctan {\dfrac {x + a} {\sqrt {b - a^2} } } + C$

where $b > a^2$.