Primitive of Reciprocal of x by a x + b squared/Proof 1
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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
\(\ds \int \frac {\d x} {x \paren {a x + b}^2}\) | \(=\) | \(\ds \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x\) | 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 |
$\blacksquare$