Primitive of Reciprocal of x cubed by a x + b
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Theorem
- $\ds \int \frac {\d x} {x^3 \paren {a x + b} } = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C$
Proof 1
\(\ds \int \frac {\d x} {x^3 \paren {a x + b} }\) | \(=\) | \(\ds \int \paren {\frac {a^2} {b^3 x} + \frac {-a} {b^2 x^2} + \frac 1 {b x^3} + \frac {-a^3} {b^3 \paren {a x + b} } } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2} {b^3} \int \frac {\d x} x + \frac {-a} {b^2} \int \frac {\d x} {x^2} + \frac 1 b \int \frac {\d x} {x^3} + \frac {-a^3} {b^3} \int \frac {\d x} {a x + b}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2} {b^3} \int \frac {\d x} x + \frac {-a} {b^2} \frac {-1} x + \frac 1 b \frac {-1} {2 x^2} + \frac {-a^3} {b^3} \int \frac {\d x} {a x + b} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2} {b^3} \ln \size x + \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {-a^3} {b^3} \int \frac {\d x} {a x + b} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2} {b^3} \ln \size x + \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {-a^3} {b^3} \frac 1 a \ln \size {a x + b} + C\) | Primitive of $\dfrac 1 {a x + b}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) | Difference of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) | rearranging |
$\blacksquare$
Proof 2
\(\ds \int \frac {\d x} {x^3 \paren {a x + b} }\) | \(=\) | \(\ds \int \frac {b \rd x} {b x^3 \paren {a x + b} }\) | multiplying top and bottom by $b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {a x + b - a x} \rd x} {b x^3 \paren {a x + b} }\) | adding and subtracting $a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 b \int \frac {\paren {a x + b} \rd x} {x^3 \paren {a x + b} } - \frac a b \int \frac {x \rd x} {x^3 \paren {a x + b} }\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 b \int \frac {\d x} {x^3} - \frac a b \int \frac {\d x} {x^2 \paren {a x + b} }\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {2 b x^2} - \frac a b \int \frac {\d x} {x^2 \paren {a x + b} } + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {2 b x^2} - \frac a b \paren {-\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} } + C\) | Primitive of $\dfrac 1 {x^2 \paren {a x + b} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {2 b x^2} + \frac a {b^2 x} - \frac {a^2} {b^3} \ln \size {\frac {a x + b} x} + C\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) | Difference of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) | rearranging |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.65$