Primitive of x squared over Root of a x squared plus b x plus c
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} } = \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
Proof
First:
\(\ds z\) | \(=\) | \(\ds a x^2 + b x + c\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds 2 a x + b\) | Derivative of Power |
Then:
\(\ds \) | \(\) | \(\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {a x^2 \rd x} {\sqrt {a x^2 + b x + c} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {\sqrt {a x^2 + b x + c} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\sqrt {a x^2 + b x + c} } - \frac b a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \sqrt {a x^2 + b x + c} \rd x - \frac b a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \paren {\frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } }\) | Primitive of $\sqrt {a x^2 + b x + c}$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac b a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \paren {\frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac b a \paren {\frac {\sqrt {a x^2 + b x + c} } a - \frac b {2 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | Primitive of $\dfrac x {\sqrt {a x^2 + b x + c} }$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 a c - b^2} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | simplifying | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {b \sqrt {a x^2 + b x + c} } {a^2} + \frac {b^2} {2 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 a c - b^2} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | common denominators | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {4 b \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 b^2} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } - \frac {8 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2 a x + b - 4 b} \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 a c - b^2 + 4 b^2 - 8 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | combining like terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.282$