Primitive of x squared over a x + b squared by p x + q
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Theorem
- $\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac {b^2} {\paren {b p - a q} a^2 \paren {a x + b} } + \frac 1 {\paren {b p - a q}^2} \paren {\frac {q^2} p \ln \size {p x + q} + \frac {b \paren {b p - 2 a q} } {a^2} \ln \size {a x + b} } + C$
Proof
\(\ds \) | \(\) | \(\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {\frac {-b^2} {a \paren {b p - a q} \paren {a x + b}^2} + \frac {q^2} {\paren {b p - a q}^2 \paren {p x + q} } + \frac {b \paren {b p - 2 a q} } {a \paren {b p - a q}^2 \paren {a x + b} } } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-b^2} {a \paren {b p - a q} } \int \frac {\d x} {\paren {a x + b}^2} + \frac {q^2} {\paren {b p - a q}^2} \int \frac {\d x} {p x + q} + \frac {b \paren {b p - 2 a q} } {a \paren {b p - a q}^2} \int \frac {\d x} {a x + b}\) | Linear Combination of Primitives |
We have:
\(\ds \int \frac {\d x} {\paren {a x + b}^2}\) | \(=\) | \(\ds \frac {-1} {a \paren {a x + b} }\) | Primitive of $\dfrac 1 {\paren {a x + b}^2}$ | |||||||||||
\(\ds \int \frac {\d x} {p x + q}\) | \(=\) | \(\ds \frac 1 p \ln \size {p x + q}\) | Primitive of $\dfrac 1 {a x + b}$ | |||||||||||
\(\ds \int \frac {\d x} {a x + b}\) | \(=\) | \(\ds \frac 1 a \ln \size {a x + b}\) | Primitive of $\dfrac 1 {a x + b}$ |
Therefore:
\(\ds \) | \(\) | \(\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-b^2} {a \paren {b p - a q} } \frac {-1} {a \paren {a x + b} } + \frac {q^2} {\paren {b p - a q}^2} \paren {\frac 1 p \ln \size {p x + q} } + \frac {b \paren {b p - 2 a q} } {a \paren {b p - a q}^2} \paren {\frac 1 a \ln \size {a x + b} } + C\) | Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {b^2} {\paren {b p - a q} a^2 \paren {a x + b} } + \frac 1 {\paren {b p - a q}^2} \paren {\frac {q^2} p \ln \size {p x + q} + \frac {b \paren {b p - 2 a q} } {a^2} \ln \size {a x + b} } + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$ and $p x + q$: $14.109$