Primitive of x squared over square of a x squared plus b x plus c
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2} = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$
Proof
\(\ds \) | \(\) | \(\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {a x^2 \rd x} {\paren {a x^2 + b x + c}^2}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {a x^2 + b x + c}^2} - \frac 1 a \int \frac {\paren {b x + c} \rd x} {\paren {a x^2 + b x + c}^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {a x^2 + b x + c}^2} - \frac b a \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} - \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d x} {a x^2 + b x + c} - \frac b a \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} - \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds - \frac b a \paren {\frac {- \paren {b x + 2 c} } {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac b {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c} }\) | Primitive of $\dfrac x {\paren {a x^2 + b x + c}^2}$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 a \int \frac {\d x} {a x^2 + b x + c}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {b^2} {a \paren {4 a c - b^2} } \int \frac {\d x} {a x^2 + b x + c}\) | multiplying out | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 a \int \frac {\d x} {a x^2 + b x + c}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {b^2 + 4 a c - b^2} {a \paren {4 a c - b^2} } \int \frac {\d x} {a x^2 + b x + c}\) | collecting terms in $\ds \int \frac {\d x} {a x^2 + b x + c}$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) | simplifying | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac c a \paren {\frac {2 a x + b} {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 a} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c} }\) | Primitive of $\dfrac 1 {\paren {a x^2 + b x + c}^2}$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {2 a c x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) | multiplying out | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x^2 + b x + c$: $14.274$