Primitive of x by Arccosecant of x over a

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Theorem

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


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:

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


and let:

\(\displaystyle \frac {\d v} {\d x}\) \(=\) \(\displaystyle x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {x^2} 2\) Primitive of Power


First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.

Then:

\(\displaystyle \int x \arccsc \frac x a \rd x\) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a - \int \frac {x^2} 2 \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac a 2 \int \frac {x \ \mathrm d x} {\sqrt {x^2 - a^2} } + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac a 2 \sqrt {x^2 - a^2} + C\) Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac {a \sqrt{x^2 - a^2} } 2 + C\) simplifying


Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$.

Then:

\(\displaystyle \int x \arccsc \frac x a \rd x\) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a - \int \frac {x^2} 2 \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac a 2 \int \frac {x \rd x} {\sqrt {x^2 - a^2} } + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac a 2 \sqrt {x^2 - a^2} + C\) Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac {a \sqrt{x^2 - a^2} } 2 + C\) simplifying

$\blacksquare$


Also see


Sources