Primitive of Reciprocal of x squared by a x + b/Proof 1
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Theorem
- $\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$
Proof
\(\ds \int \frac {\d x} {x^2 \paren {a x + b} }\) | \(=\) | \(\ds \int \paren {-\frac a {b^2 x} + \frac 1 {b x^2} + \frac {a^2} {b^2 \paren {a x + b} } } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac a {b^2} \int \frac {\d x} x + \frac 1 b \int \frac {\d x} {x^2} + \frac {a^2} {b^2} \int \frac {\d x} {a x + b}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac a {b^2} \int \frac {\d x} x + \frac 1 b \frac {-1} x + \frac {a^2} {b^2} \int \frac {\d x} {a x + b} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac a {b^2} \ln \size x - \frac 1 {b x} + \frac {a^2} {b^2} \int \frac {\d x} {a x + b} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac a {b^2} \ln \size x - \frac 1 {b x} + \frac a {b^2} \ln \size {a x + b} + C\) | Primitive of $\dfrac 1 {a x + b}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C\) | Difference of Logarithms |
$\blacksquare$