Primitive of Inverse Hyperbolic Secant of x over a

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Theorem

$\displaystyle \int \operatorname{sech}^{-1} \frac x a \ \mathrm d x = \begin{cases} x \operatorname{sech}^{-1} \dfrac x a + a \arcsin \dfrac x a + C & : \operatorname{sech}^{-1} \dfrac x a > 0 \\ x \operatorname{sech}^{-1} \dfrac x a - a \arcsin \dfrac x a + C & : \operatorname{sech}^{-1} \dfrac x a < 0 \end{cases}$


Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

\(\displaystyle u\) \(=\) \(\displaystyle \operatorname{sech}^{-1} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d u} {\mathrm d x}\) \(=\) \(\displaystyle \frac {-a} {x \sqrt{a^2 - x^2} }\) Derivative of $\operatorname{sech}^{-1} \dfrac x a$


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle x\) Primitive of Constant


Then:

\(\displaystyle \int \operatorname{sech}^{-1} \frac x a \ \mathrm d x\) \(=\) \(\displaystyle x \operatorname{sech}^{-1} \frac x a - \int x \left({\frac {-a} {x \sqrt{a^2 - x^2} } }\right) \ \mathrm d x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle x \operatorname{sech}^{-1} \frac x a + a \int \frac {\mathrm d x} {\sqrt{a^2 - x^2} } + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle x \operatorname{sech}^{-1} \frac x a + \arcsin \frac x a + C\) Primitive of $\dfrac 1 {\sqrt{a^2 - x^2} }$



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