Primitive of Reciprocal of x by a x squared plus b x plus c
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int \frac {\d x} {x \paren {a x^2 + b x + c} } = \frac 1 {2 c} \ln \size {\frac {x^2} {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c}$
Proof
\(\ds \int \frac {\d x} {x \paren {a x^2 + b x + c} }\) | \(=\) | \(\ds \int \paren {\frac 1 {c x} - \frac {a x + b} {c \paren {a x^2 + b x + c} } } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \int \frac {\d x} x - \frac a c \int \frac {x \rd x} {a x^2 + b x + c} - \frac b c \int \frac {\d x} {a x^2 + b x + c}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \ln \size x - \frac a c \int \frac {x \rd x} {a x^2 + b x + c} - \frac b c \int \frac {\d x} {a x^2 + b x + c}\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \ln \size x - \frac a c \paren {\frac 1 {2 a} \ln \size {a x^2 + b x + c} - \frac b {2 a} \int \frac {\d x} {a x^2 + b x + c} } - \frac b c \int \frac {\d x} {a x^2 + b x + c}\) | Primitive of $\dfrac x {a x^2 + b x + c}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \ln \size x - \frac 1 {2 c} \ln \size {a x^2 + b x + c} + \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c} - \frac b c \int \frac {\d x} {a x^2 + b x + c}\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \ln \size x - \frac 1 {2 c} \ln \size {a x^2 + b x + c} - \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 c} \ln \size {x^2} - \frac 1 {2 c} \ln \size {a x^2 + b x + c} - \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c}\) | Logarithm of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 c} \ln \size {\frac {x^2} {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c}\) | Difference of Logarithms |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x^2 + bx + c$: $14.269$