Primitive of Reciprocal of x cubed by a x + b squared
Jump to navigation
Jump to search
Theorem
- $\ds \int \frac {\d x} {x^3 \paren {a x + b}^2} = - \frac {\paren {a x + b}^2} {2 b^4 x^2} + \frac {3 a \paren {a x + b} } {b^4 x} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C$
Proof
\(\ds \int \frac {\d x} {x^3 \paren {a x + b}^2}\) | \(=\) | \(\ds \int \paren {\frac {3 a^2} {b^4 x} + \frac {-2 a} {b^3 x^2} + \frac 1 {b^2 x^3} + \frac {-3 a^3} {b^4 \paren {a x + b} } + \frac {-a^3} {b^3 \paren {a x + b}^2} } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 a^2} {b^4} \int \frac {\d x} x + \frac {-2 a} {b^3} \int \frac {\d x} {x^2} + \frac 1 {b^2} \int \frac {\d x} {x^3} + \frac {-3 a^3} {b^4} \int \frac {\d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 a^2} {b^4} \int \frac {\d x} x + \frac {-2 a} {b^3} \frac {-1} x + \frac 1 {b^2} \frac {-1} {2 x^2} + \frac {-3 a^3} {b^4} \int \frac {\d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 a^2} {b^4} \ln \size x + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^3} {b^4} \int \frac {\d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 a^2} {b^4} \ln \size x + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^3} {b^4} \frac 1 a \ln \size {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + C\) | Primitive of Reciprocal of a x + b | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 a^2} {b^4} \ln \size x + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^2} {b^4} \ln \size {a x + b} + \frac {-a^3} {b^3} \frac {-1} {a \paren {a x + b} } + C\) | Primitive of Reciprocal of a x + b squared | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {a^2} {b^3 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) | Difference of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {a^2} {b^3 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + \frac {3 a^2} {2 b^4} + C\) | C is an arbitrary constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 a b \paren {a x + b} x} {2 b^4 \paren {a x + b} x^2} - \frac {b^2 \paren {a x + b} } {2 b^4 \paren {a x + b} x^2} + \frac {2 a^2 b x^2} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + \frac {3 a^2 \paren {a x + b} x^2} {2 b^4 \paren {a x + b} x^2} + C\) | Represent with common denominators | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 a b \paren {a x + b} x - b^2 \paren {a x + b} + 2 a^2 b x^2 + 3 a^2 \paren {a x + b} x^2} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) | Combine fractions | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 a^2 b x^2 + 4 a b^2 x - a b^2 x - b^3 + 2 a^2 b x^2 + 3 a^3 x^3 + 3 a^2 b x^2} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) | Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {6 a \paren {a x + b}^2 x - \paren {a x + b}^3 - 2 a^3 x^3} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) | Factorisation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {6 a \paren {a x + b}^2 x} {2 b^4 \paren {a x + b} x^2} - \frac {\paren {a x + b}^3} {2 b^4 \paren {a x + b} x^2} - \frac {2 a^3 x^3} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) | Split into separate fractions | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 a \paren {a x + b} } {b^4 x} - \frac {\paren {a x + b}^2} {2 b^4 x^2} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) | Simplify |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.72$