Primitive of Reciprocal of a x squared plus b x plus c/Zero Discriminant
Jump to navigation
Jump to search
Theorem
Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c = 0$.
Then:
- $\ds \int \frac {\d x} {a x^2 + b x + c} = \frac {-2} {2 a x + b} + C$
Proof
First:
\(\ds a x^2 + b x + c\) | \(=\) | \(\ds \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}\) | Completing the Square | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {a x^2 + b x + c}\) | \(=\) | \(\ds \int \frac {4 a \rd x} {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} }\) |
Let $b^2 - 4 a c = 0$.
Then:
\(\ds \int \frac {\d x} {a x^2 + b x + c}\) | \(=\) | \(\ds \int \frac {4 a \rd x} {\paren {2 a x + b}^2}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-4 a} {2 a \paren {2 a x + b} } + C\) | Primitive of $\dfrac 1 {\paren {a x + b}^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-2} {2 a x + b} + C\) | simplifying |
$\blacksquare$
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Rational Algebraic Functions: $3.3.18$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x^2 + b x + c$: $14.265$