Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form

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Theorem

$\displaystyle \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$


Proof

Let:

\(\displaystyle x\) \(=\) \(\displaystyle a \sinh \theta\)
\(\text {(1)}: \quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d x} {\d \theta}\) \(=\) \(\displaystyle a \cosh \theta\) Derivative of Hyperbolic Sine


Also:

\(\displaystyle x\) \(=\) \(\displaystyle a \sinh \theta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2 + a^2\) \(=\) \(\displaystyle a^2 \sinh^2 \theta + a^2\)
\(\displaystyle \) \(=\) \(\displaystyle a^2 \paren {\sinh^2 \theta + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle a^2 \cosh^2 \theta\) Difference of Squares of Hyperbolic Cosine and Sine
\(\text {(2)}: \quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle \sqrt {x^2 + a^2}\) \(=\) \(\displaystyle a \cosh \theta\)


and:

\(\displaystyle x\) \(=\) \(\displaystyle a \sinh \theta\)
\(\text {(3)}: \quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle \theta\) \(=\) \(\displaystyle \sinh^{-1} \frac x a\) Definition 1 of Real Inverse Hyperbolic Sine


Thus:

\(\displaystyle \int \sqrt {x^2 + a^2} \rd x\) \(=\) \(\displaystyle \int \sqrt {x^2 + a^2} \, a \cosh \theta \rd \theta\) Integration by Substitution from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle \int a^2 \cosh^2 \theta \rd \theta\) substituting for $\sqrt {x^2 + a^2}$ from $(2)$
\(\displaystyle \) \(=\) \(\displaystyle a^2 \int \cosh^2 \theta \rd \theta\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle a^2 \frac {\sinh \theta \cosh \theta + \theta} 2 + C\) Primitive of Square of Hyperbolic Cosine Function: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 a \sinh \theta a \cosh \theta + \frac {a^2 \theta} 2 + C\) rearranging
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 x a \cosh \theta + \frac {a^2 \theta} 2 + C\) substituting $x = a \sinh \theta$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 x \sqrt {x^2 + a^2} + \frac {a^2 \theta} 2 + C\) substituting $\sqrt {x^2 + a^2} = a \cosh \theta$ from $(2)$
\(\displaystyle \) \(=\) \(\displaystyle \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C\) substituting for $\theta$ from $(3)$

$\blacksquare$