Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form
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Theorem
- $\ds \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$
Proof
Let:
\(\ds x\) | \(=\) | \(\ds a \sinh \theta\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d \theta}\) | \(=\) | \(\ds a \cosh \theta\) | Derivative of Hyperbolic Sine |
Also:
\(\ds x\) | \(=\) | \(\ds a \sinh \theta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 + a^2\) | \(=\) | \(\ds a^2 \sinh^2 \theta + a^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \paren {\sinh^2 \theta + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \cosh^2 \theta\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \sqrt {x^2 + a^2}\) | \(=\) | \(\ds a \cosh \theta\) |
and:
\(\ds x\) | \(=\) | \(\ds a \sinh \theta\) | ||||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \theta\) | \(=\) | \(\ds \sinh^{-1} \frac x a\) | Definition 1 of Real Inverse Hyperbolic Sine |
Thus:
\(\ds \int \sqrt {x^2 + a^2} \rd x\) | \(=\) | \(\ds \int \sqrt {x^2 + a^2} \, a \cosh \theta \rd \theta\) | Integration by Substitution from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int a^2 \cosh^2 \theta \rd \theta\) | substituting for $\sqrt {x^2 + a^2}$ from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \int \cosh^2 \theta \rd \theta\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \frac {\sinh \theta \cosh \theta + \theta} 2 + C\) | Primitive of Square of Hyperbolic Cosine Function: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 a \sinh \theta a \cosh \theta + \frac {a^2 \theta} 2 + C\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 x a \cosh \theta + \frac {a^2 \theta} 2 + C\) | substituting $x = a \sinh \theta$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 x \sqrt {x^2 + a^2} + \frac {a^2 \theta} 2 + C\) | substituting $\sqrt {x^2 + a^2} = a \cosh \theta$ from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C\) | substituting for $\theta$ from $(3)$ |
$\blacksquare$
Sources
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $21$.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals