Primitive of Reciprocal of x by a x + b

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Theorem

$\ds \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$


Proof 1

\(\ds \int \frac {\d x} {x \paren {a x + b} }\) \(=\) \(\ds \int \paren {\dfrac 1 {b x} - \dfrac a {b \paren {a x + b} } } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac 1 b \int \frac {\d x} x - \frac a b \int \frac {\d x} {a x + b}\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \frac 1 b \ln \size x - \frac a b \int \frac {\d x} {a x + b} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac 1 b \ln \size x - \frac 1 b \ln \size {a x + b} + C\) Primitive of $\dfrac 1 {a x + b}$
\(\ds \) \(=\) \(\ds \frac 1 b \ln \size {\frac x {a x + b} } + C\) Difference of Logarithms

$\blacksquare$


Proof 2

\(\ds \int \frac {\d x} {x \paren {a x + b} }\) \(=\) \(\ds \int \frac {b \rd x} {b x \paren {a x + b} }\) multiplying top and bottom by $b$
\(\ds \) \(=\) \(\ds \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b} }\) adding and subtracting $a x$
\(\ds \) \(=\) \(\ds \frac 1 b \int \frac {\paren {a x + b} \rd x} {x \paren {a x + b} } - \frac a b \int \frac {x \rd x} {x \paren {a x + b} }\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \frac 1 b \int \frac {\d x} x - \frac a b \int \frac {\d x} {a x + b}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 b \ln \size x - \frac a b \int \frac {\d x} {a x + b} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac 1 b \ln \size x - \frac 1 b \ln \size {a x + b} + C\) Primitive of $\dfrac 1 {a x + b}$
\(\ds \) \(=\) \(\ds \frac 1 b \ln \size {\frac x {a x + b} } + C\) Difference of Logarithms

$\blacksquare$


Sources