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

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

$\ds \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} } = \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {10} } + C$

Proof

\(\ds \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }\)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac {10} 9 - \paren {x - \frac 2 3}^2} }\)

Completing the Square

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {x - \frac 2 3} {\sqrt {\frac {10} 9} } } + C\)

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

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C\)

simplifying

$\blacksquare$

Sources

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