Primitive of Reciprocal of x squared by a x squared plus b x plus c

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Theorem

Let $a \in \R_{\ne 0}$.

Then:

$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} } = \frac b {2 c^2} \ln \size {\frac {a x^2 + b x + c} {x^2} } - \frac 1 {c x} + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}$


Proof

\(\ds \) \(\) \(\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \int \paren {\frac {-b} {c^2 x} + \frac 1 {c x^2} + \frac {a b x + b^2 - a c} {c^2 \paren {a x^2 + b x + c} } } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac {-b} {c^2} \int \frac {\d x} x + \frac 1 c \int \frac {\d x} {x^2} + \frac {a b} {c^2} \int \frac {x \rd x} {a x^2 + b x + c} + \frac {b^2 - a c} {c^2} \int \frac {\d x} {a x^2 + b x + c}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {-b} {c^2} \ln \size x + \frac 1 c \int \frac {\d x} {x^2} + \frac {a b} {c^2} \int \frac {x \rd x} {a x^2 + b x + c} + \frac {b^2 - a c} {c^2} \int \frac {\d x} {a x^2 + b x + c}\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac {-b} {c^2} \ln \size x - \frac 1 {c x} + \frac {a b} {c^2} \int \frac {x \rd x} {a x^2 + b x + c} + \frac {b^2 - a c} {c^2} \int \frac {\d x} {a x^2 + b x + c}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-b} {c^2} \ln \size x - \frac 1 {c x} + \frac {a b} {c^2} \paren {\frac 1 {2 a} \ln \size {a x^2 + b x + c} - \frac b {2 a} \int \frac {\d x} {a x^2 + b x + c} } + \frac {b^2 - a c} {c^2} \int \frac {\d x} {a x^2 + b x + c}\) Primitive of $\dfrac x {a x^2 + b x + c}$
\(\ds \) \(=\) \(\ds \frac {-b} {c^2} \ln \size x - \frac 1 {c x} + \frac b {2 c^2} \ln \size {a x^2 + b x + c} - \frac {b^2} {2 c^2} \int \frac {\d x} {a x^2 + b x + c} + \frac {2 b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}\) multiplying out
\(\ds \) \(=\) \(\ds \frac {-b} {c^2} \ln \size x - \frac 1 {c x} + \frac b {2 c^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}\) simplifying
\(\ds \) \(=\) \(\ds \frac {-b} {2 c^2} \ln \size {x^2} - \frac 1 {c x} + \frac b {2 c^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}\) Logarithm of Power
\(\ds \) \(=\) \(\ds \frac b {2 c^2} \ln \size {\frac {a x^2 + b x + c} {x^2} } - \frac 1 {c x} + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}\) Difference of Logarithms

$\blacksquare$


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