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