Primitive of Reciprocal of x by a x + b squared
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Theorem
- $\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$
Proof 1
| \(\ds \int \frac {\d x} {x \paren {a x + b}^2}\) | \(=\) | \(\ds \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x\) | Partial Fraction Expansion | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b^2} \int \frac {\d x} x - \frac a {b^2} \int \frac {\d x} {a x + b} - \frac a b \int \frac {\d x} {\paren {a x + b}^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b^2} \ln \size x - \frac a {b^2} \int \frac {\d x} {a x + b} - \frac a b \int \frac {\d x} {\paren {a x + b}^2} + C\) | Primitive of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b^2} \ln \size x - \frac a {b^2} \ln \size {a x + b} - \frac a b \int \frac {\d x} {\paren {a x + b}^2} + C\) | Primitive of $\dfrac 1 {a x + b}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b^2} \ln \size x - \frac a {b^2} \ln \size {a x + b} - \frac a b \frac {-1} {a \paren {a x + b} } + C\) | Primitive of $\dfrac 1 {\paren {a x + b}^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C\) | Difference of Logarithms and rearranging |
$\blacksquare$
Proof 2
| \(\ds \int \frac {\d x} {x \paren {a x + b}^2}\) | \(=\) | \(\ds \int \frac {b \rd x} {b x \paren {a x + b}^2}\) | multiplying top and bottom by $b$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}\) | adding and subtracting $a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 b \int \frac {\paren {a x + b} \rd x} {x \paren {a x + b}^2} - \frac a b \int \frac {x \rd x} {x \paren {a x + b}^2}\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 b \int \frac {\d x} {x \paren {a x + b} } - \frac a b \int \frac {\d x} {\paren {a x + b}^2}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 b \paren {\frac 1 b \ln \size {\frac x {a x + b} } } - \frac a b \int \frac {\d x} {\paren {a x + b}^2} + C\) | Primitive of $\dfrac 1 {x \paren {a x + b} }$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b^2} \ln \size {\frac x {a x + b} } - \frac a b \paren {-\frac 1 {a \paren {a x + b} } } + C\) | Primitive of $\dfrac 1 {\paren {a x + b}^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.70$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.9.$