Primitive of Root of a x + b over x
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Theorem
- $\ds \int \frac {\sqrt {a x + b} } x \rd x = 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt{a x + b} }$
Proof 1
- $\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Putting $m = -1$ and $n = \dfrac 1 2$:
\(\ds \int \frac {\sqrt {a x + b} } x \rd x\) | \(=\) | \(\ds \int x^{-1} \paren {a x + b}^{1/2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^0 \paren {a x + b}^{1/2} } {\frac 1 2} + \frac {\frac 1 2 b} {\frac 1 2} \int x^{-1} \paren {a x + b}^{- 1/2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt {a x + b} }\) | simplifying |
$\blacksquare$
Proof 2
Let:
\(\ds v\) | \(=\) | \(\ds \sqrt x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \frac 1 {2 \sqrt x}\) | Power Rule for Derivatives | ||||||||||
\(\ds u\) | \(=\) | \(\ds \frac {2 \sqrt {a x + b} } {\sqrt x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {\frac {\sqrt x \cdot 2 a} {2 \sqrt{a x + b} } - \frac {2 \sqrt {a x + b} } {2 \sqrt x} } x\) | Quotient Rule for Derivatives etc. | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-b} {x^{3/2} \sqrt {a x + b} }\) | simplifying |
From Integration by Parts:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \, \frac {\d u} {\d x} \rd x$
from which:
\(\ds \int \frac {\sqrt {a x + b} } x \rd x\) | \(=\) | \(\ds \int \frac {2 \sqrt {a x + b} }{\sqrt x} \frac 1 {2 \sqrt x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sqrt {a x + b} } {\sqrt x} \sqrt x - \int {\sqrt x} \frac {-b} {x^{3/2} \sqrt {a x + b} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt {a x + b} }\) | simplification |
$\blacksquare$
Also see
- Primitive of Reciprocal of $x \sqrt {a x + b}$ for $\ds \int \frac {\d x} {x \sqrt {a x + b} }$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.92$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $12$.