Primitive of x over a x + b squared by p x + q
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Theorem
- $\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C$
Corollary
- $\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } + \frac x {a x + b} } + C$
Proof
| \(\ds \) | \(\) | \(\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \paren {\frac 1 {b p - a q} \paren {\frac {a q} {\paren {b p - a q} \paren {a x + b} } + \frac b {\paren {a x + b}^2} - \frac {p q} {\paren {b p - a q} \paren {p x + q} } } } \rd x\) | Partial Fraction Expansion | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b p - a q} \paren {\frac {a q} {b p - a q} \int \frac {\d x} {a x + b} + b \int \frac {\d x} {\paren {a x + b}^2} - \frac {p q} {b p - a q} \int \frac {\d x} {p x + q} }\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b p - a q} \paren {\frac {a q} {b p - a q} \frac 1 a \ln \size {a x + b} + b \int \frac {\d x} {\paren {a x + b}^2} + \frac {p q} {b p - a q} \frac 1 p \ln \size {p x + q} } + C\) | Primitive of $\dfrac 1 {a x + b}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {a x + b} + b \frac {-1} {a \paren {a x + b} } - \frac q {b p - a q} \ln \size {p x + q} } + C\) | Primitive of $\dfrac 1 {\paren {a x + b}^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C\) | Difference of Logarithms |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$ and $p x + q$: $14.108$