Primitive of Root of a x squared plus b x plus c over x
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int \frac {\sqrt {a x^2 + b x + c} } x \rd x = \sqrt {a x^2 + b x + c} + \frac b 2 \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$
Proof
\(\ds \) | \(\) | \(\ds \int \frac {\sqrt {a x^2 + b x + c} } x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {a x^2 + b x + c} {x \sqrt {a x^2 + b x + c} } \rd x\) | multiplying top and bottom by $\sqrt {a x^2 + b x + c}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a \int \frac {x^2 \rd x} {x \sqrt {a x^2 + b x + c} } + b \int \frac {x \rd x} {x \sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } + b \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds a \paren {\frac {\sqrt {a x^2 + b x + c} } a - \frac b {2 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } }\) | Primitive of $\dfrac x {\sqrt {a x^2 + b x + c} }$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds b \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {a x^2 + b x + c} + \frac b 2 \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x^2 + bx + c}$: $14.288$