Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Negative Discriminant

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $a \in \R_{>0}$.

Let $b^2 - 4 a c < 0$.

Then for $x \in \R$ such that $a x^2 + b x + c > 0$:

$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$

where $\arsinh$ denotes the area hyperbolic sine function.


Proof

Completing the Square

First:

\(\ds a x^2 + b x + c\) \(=\) \(\ds \frac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}\) Completing the Square
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) \(=\) \(\ds \int \frac {2 \sqrt a \rd x} {\sqrt {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } }\)


Put:

\(\ds z\) \(=\) \(\ds 2 a x + b\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds 2 a\) Derivative of Power


Let $D = b^2 - 4 a c$.

Thus:

\(\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) \(=\) \(\ds \int \frac {2 \sqrt a \rd x} {\sqrt {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } }\) from $(1)$
\(\ds \) \(=\) \(\ds \int \frac {\d z} {\sqrt a \sqrt {z^2 - D} }\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt a} \int \frac {\d z} {\sqrt {z^2 - D} }\) Primitive of Constant Multiple of Function


Let $b^2 - 4 a c < 0$.

Then:

\(\ds - D\) \(>\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds -D\) \(=\) \(\ds q^2\) for some $q \in \R$
\(\ds \leadsto \ \ \) \(\ds q\) \(=\) \(\ds \sqrt {4 a c - b^2}\) by definition of $D$


Thus:

\(\ds \int \frac {\d x} {\sqrt{a x^2 + b x + c} }\) \(=\) \(\ds \frac 1 {\sqrt a} \int \frac {\d z} {\sqrt {z^2 + q^2} }\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt a} \map \arsinh {\frac z q} + C\) Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt a} \map \arsinh {\frac {2 a x + b} {\sqrt {4 a c - b^2} } } + C\) substituting for $z$ and $q$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_greater_than_0/Negative_Discriminant&oldid=630034"