Primitive of Reciprocal of x squared by a x + b cubed
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Theorem
- $\ds \int \frac {\d x} {x^2 \paren {a x + b}^3} = \frac {-a} {2 b^2 \paren {a x + b}^2} - \frac {2 a} {b^3 \paren {a x + b} } - \frac 1 {b^3 x} + \frac {3 a} {b^4} \ln \size {\frac {a x + b} x} + C$
Proof
\(\ds \int \frac {\d x} {x^2 \paren {a x + b}^3}\) | \(=\) | \(\ds \int \paren {\frac {-3 a} {b^4 x} + \frac 1 {b^3 x^2} + \frac {3 a^2} {b^4 \paren {a x + b} } + \frac {2 a^2} {b^3 \paren {a x + b}\^2} + \frac {a^2} {b^2 \paren {a x + b}^3} } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-3 a} {b^4} \int \frac {\d x} x + \frac 1 {b^3} \int \frac {\d x} {x^2} + \frac {3 a^2} {b^4} \int \frac {\d x} {\paren {a x + b} } + \frac {2 a^2} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^3}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-3 a} {b^4} \ln \size x + \frac 1 {b^3} \int \frac {\d x} {x^2} + \frac {3 a^2} {b^4} \int \frac {\d x} {\paren {a x + b} } + \frac {2 a^2} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^3} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-3 a} {b^4} \ln \size x + \frac 1 {b^3} \int \frac {\d x} {x^2} + \frac {3 a^2} {b^4} \ln \size {a x + b} + \frac {2 a^2} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^3} + C\) | Primitive of Reciprocal of a x + b | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-3 a} {b^4} \ln \size x + \frac 1 {b^3} \int \frac {\d x} {x^2} + \frac {3 a^2} {b^4} \ln \size {a x + b} + \frac {2 a^2} {b^3} \frac {-1} {a \paren {a x + b} } + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^3} + C\) | Primitive of Reciprocal of a x + b squared | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-3 a} {b^4} \ln \size x + \frac 1 {b^3} \int \frac {\d x} {x^2} + \frac {3 a^2} {b^4} \ln \size {a x + b} - \frac {2 a} {b^3 \paren {a x + b} } + \frac {a^2} {b^2} \frac {-1} {2 a \paren {a x + b}^2} + C\) | Primitive of Reciprocal of a x + b cubed | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-3 a} {b^4} \ln \size x + \frac 1 {b^3} \frac {-1} x + \frac {3 a^2} {b^4} \ln \size {a x + b} - \frac {2 a} {b^3 \paren {a x + b} } - \frac a {2 b^2 \paren {a x + b}^2} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a} {2 b^2 \paren {a x + b}^2} - \frac {2 a} {b^3 \paren {a x + b} } - \frac 1 {b^3 x} + \frac {3 a} {b^4} \ln \size {\frac {a x + b} x} + C\) | Difference of Logarithms and rearranging |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.78$