Primitive of Power of x over Power of a x squared plus b x plus c

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Theorem

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

Then:

\(\ds \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}\) \(=\) \(\ds \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {\paren {n - m} b} {\paren {2 n - m - 1} a} \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n}\)


Proof

With a view to expressing the problem in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \frac 1 {\paren {a x^2 + b x + c}^n}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-\paren {2 a x + b} n} {\paren {a x^2 + b x + c}^{n + 1} }\) Derivative of Power and Chain Rule for Derivatives


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds x^m\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {x^{m + 1} } {m + 1}\) Primitive of Power


Then:

\(\ds \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}\) \(=\) \(\ds \int \frac 1 {\paren {a x^2 + b x + c}^n} x^m \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 {\paren {a x^2 + b x + c}^n} \frac {x^{m + 1} } {m + 1} - \int \frac {x^{m + 1} } {m + 1} \frac {-\paren {2 a x + b} n} {\paren {a x^2 + b x + c}^{n + 1} } \rd x\) Integration by Parts
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \frac {x^{m + 1} } {\paren {m + 1} \paren {a x^2 + b x + c}^n} + \frac n {m + 1} \int \frac {\paren {2 a x + b} x^{m + 1} } {\paren {a x^2 + b x + c}^{n + 1} } \rd x\) simplifying


Meanwhile:

\(\ds \) \(\) \(\ds \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {x^{m - 2} a x^2 \rd x} {\paren {a x^2 + b x + c}^n}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {x^{m - 2} \paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^{n - 1} }\) Linear Combination of Primitives
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac b a \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n} - \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}\) and simplification
\(\ds \) \(=\) \(\ds \frac 1 a \paren {\frac {x^{m - 1} } {\paren {m - 1} \paren {a x^2 + b x + c}^{n - 1} } + \frac {n - 1} {m - 1} \int \frac {\paren {2 a x + b} x^{m - 1} } {\paren {a x^2 + b x + c}^n} \rd x}\) from $(1)$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac b a \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n} - \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}\) and simplification
\(\ds \) \(=\) \(\ds \frac {x^{m - 1} } {a \paren {m - 1} \paren {a x^2 + b x + c}^{n - 1} } + \frac {2 \paren {n - 1} } {m - 1} \int \frac {x^m} {\paren {a x^2 + b x + c}^n} \rd x\) multiplying out
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac {b \paren {n - 1} } {a \paren {m - 1} } \int \frac {x^{m - 1} } {\paren {a x^2 + b x + c}^n} \rd x\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac b a \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n} - \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\ds \paren {1 - \frac {2 \paren {n - 1} } {m - 1} } \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}\) rearranging
\(\ds \) \(=\) \(\ds \frac {x^{m - 1} } {a \paren {m - 1} \paren {a x^2 + b x + c}^{n - 1} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\frac {b \paren {n - 1} } {a \paren {m - 1} } - \frac b a} \int \frac {x^{m - 1} } {\paren {a x^2 + b x + c}^n} \rd x\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\ds \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(=\) \(\ds \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {\paren {n - m} b} {\paren {2 n - m - 1} a} \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n}\) simplification

$\blacksquare$


Sources

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