Primitive of Arccosecant of x over a

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Theorem

$\displaystyle \int \operatorname{arccsc} \frac x a \rd x = \begin{cases} \displaystyle x \operatorname{arccsc} \frac x a + a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \displaystyle x \operatorname{arccsc} \frac x a - a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}$


Proof

With a view to expressing the primitive in the form:

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

let:

\(\displaystyle u\) \(=\) \(\displaystyle \operatorname{arccsc} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}\) Derivative of $\operatorname{arccsc} \dfrac x a$


and let:

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


First let $\operatorname{arccsc} \dfrac x a$ be in the interval $\left({0 \,.\,.\,\dfrac \pi 2}\right)$.

Then:

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


Similarly, let $\operatorname{arccsc} \dfrac x a$ be in the interval $\left({-\dfrac \pi 2 \,.\,.\, 0}\right)$.

Then:

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

$\blacksquare$


Also see


Sources