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

Theorem

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

Then:

\(\ds \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}\)

\(=\)

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

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n}\)

\(\ds \)

\(\)

\(\, \ds - \, \)

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

Proof

First:

\(\ds \)

\(\)

\(\ds \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}\)

\(\ds \)

\(=\)

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

multiplying top and bottom by $c$

\(\ds \)

\(=\)

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

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

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

adding and subtracting $a x^2 + b x$

\(\ds \)

\(=\)

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

Linear Combination of Primitives

\(\ds \)

\(\)

\(\, \ds - \, \)

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

\(\text {(1)}: \quad\)

\(\ds \)

\(=\)

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

simplification

\(\ds \)

\(\)

\(\, \ds - \, \)

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

Next, with a view to obtaining an expression 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:

\(\ds u\)

\(=\)

\(\ds \frac 1 {\paren {a x^2 + b x + c}^{n - 1} }\)

\(\ds \)

\(=\)

\(\ds \paren {a x^2 + b x + c}^{-\paren {n - 1} }\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d u} {\d x}\)

\(=\)

\(\ds -\paren {n - 1} \paren {a x^2 + b x + c}^{-\paren {n - 1} - 1} \paren {2 a x + b}\)

Chain Rule for Derivatives and Derivative of Power

\(\ds \)

\(=\)

\(\ds \frac {-\paren {n - 1} \paren {2 a x + b} } {\paren {a x^2 + b x + c}^n}\)

simplifying

and let:

\(\ds \frac {\d v} {\d x}\)

\(=\)

\(\ds \frac 1 {x^m}\)

\(\ds \)

\(=\)

\(\ds x^-m\)

\(\ds \leadsto \ \ \)

\(\ds v\)

\(=\)

\(\ds \frac {x^{-m + 1} } {-m + 1}\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac {-1} {\paren {m - 1} x^{m - 1} }\)

simplifying

Then:

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

\(=\)

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

Note the index is $n - 1$

\(\ds \)

\(=\)

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

Integration by Parts

\(\ds \)

\(\)

\(\, \ds - \, \)

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

\(\text {(2)}: \quad\)

\(\ds \)

\(=\)

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

simplification

\(\ds \)

\(\)

\(\, \ds - \, \)

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

and Linear Combination of Primitives

\(\ds \)

\(\)

\(\, \ds - \, \)

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

Thus:

\(\ds \)

\(\)

\(\ds \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}\)

\(\ds \)

\(=\)

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

$(1)$

\(\ds \)

\(=\)

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

$(2)$

\(\ds \)

\(\)

\(\, \ds - \, \)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(\)

\(\, \ds - \, \)

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

\(\ds \)

\(=\)

\(\ds \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} } - \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac {\paren {m - n + 2} b} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}\)

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x^2 + b x + c$: $14.279$