Primitive of x squared over Cube of Root of a x squared plus b x plus c
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {\paren {2 b^2 - 4 a c} x + 2 b c} {a \paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
Proof
\(\ds \) | \(\) | \(\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {a x^2 \rd x} {a \paren {\sqrt {a x^2 + b x + c} }^3}\) | multiplying top and bottom by $a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {a \paren {\sqrt {a x^2 + b x + c} }^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac b a \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac c a \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} } - \frac b a \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac c a \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac b a \paren {\frac {2 \paren {b x + 2 c} } {\paren {b^2 - 4 a c} \sqrt {a x^2 + b x + c} } }\) | Primitive of $\dfrac x {\paren {\sqrt {a x^2 + b x + c} }^3}$ | ||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac c a \paren {\frac {2 \paren {2 a x + b} } {\paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } }\) | Primitive of $\dfrac 1 {\paren {\sqrt {a x^2 + b x + c} }^3}$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2 b^2 - 4 a c} x + 2 b c} {a \paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.292$