Primitive of Reciprocal of x squared by square of a x squared plus b x plus c/Proof 2
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$
Proof
First:
\(\ds \) | \(\) | \(\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {c \rd x} {c x^2 \paren {a x^2 + b x + c}^2}\) | multiplying top and bottom by $c$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \int \frac {c \rd x} {x^2 \paren {a x^2 + b x + c}^2}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \int \frac {a x^2 + b x + c - a x^2 - b x} {x^2 \paren {a x^2 + b x + c}^2} \rd x\) | adding and subtracting $a x^2 + b x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \int \frac {\paren {a x^2 + b x + c} \rd x} {x^2 \paren {a x^2 + b x + c}^2} - \frac a c \int \frac {x^2 \rd x} {x^2 \paren {a x^2 + b x + c}^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac b c \int \frac {x \rd x} {x^2 \paren {a x^2 + b x + c}^2}\) | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac 1 c \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} } - \frac a c \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) | simplification | ||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac b c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}\) |
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:
\(\ds u\) | \(=\) | \(\ds \frac 1 {\paren {a x^2 + b x + c} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a x^2 + b x + c}^{-1}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -\paren {2 a x + b} \paren {a x^2 + b x + c}^{-2}\) | Chain Rule for Derivatives and Derivative of Power | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\paren {2 a x + b} } {\paren {a x^2 + b x + c}^2}\) | simplifying |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \frac 1 {x^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {-1} x\) | Primitive of Power |
Then:
\(\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} }\) | \(=\) | \(\ds \int \frac 1 {\paren {a x^2 + b x + c} } \frac 1 {x^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {a x^2 + b x + c} } \frac {-1} x - \int \frac {-1} x \frac {-\paren {2 a x + b} } {\paren {a x^2 + b x + c}^2} \rd x\) | Integration by Parts | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac {-1} {x \paren {a x^2 + b x + c} } - 2 a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) | simplification | ||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds b \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}\) | Linear Combination of Primitives |
Thus:
\(\ds \) | \(\) | \(\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} } - \frac a c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac b c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 c \paren {\frac {-1} {x \paren {a x^2 + b x + c} } - 2 a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - b \int \frac {\d x} {x \paren {a x^2 + b x + c}^2} }\) | from $(2)$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac a c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac b c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}\) | gathering terms |
$\blacksquare$