Primitive of Reciprocal of a x squared plus b x plus c/Examples/3 x^2 + 4 x + 2/Proof 2

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

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

Proof

\(\ds \int \frac {\d x} {3 x^2 + 4 x + 2}\)

\(=\)

\(\ds \dfrac 1 3 \int \frac {\d x} {x^2 + \frac 4 3 x + \frac 2 3}\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \paren {\frac 2 3 - \frac 4 9} }\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \frac 2 9}\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \paren {\frac {\sqrt 2} 3}^2}\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \paren {\dfrac 3 {\sqrt 2} \map \arctan {\dfrac {x + \frac 2 3} {\frac {\sqrt 2} 3} } } + C\)

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

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C\)

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration