Primitive of x by Square of Cosecant of a x

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Theorem

$\displaystyle \int x \csc^2 a x \ \mathrm d x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \left\vert{\sin a x}\right\vert + C$


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:

\(\displaystyle u\) \(=\) \(\displaystyle x\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d u} {\mathrm d x}\) \(=\) \(\displaystyle 1\) Derivative of Identity Function


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle \csc^2 a x\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {-\cot a x} a\) Primitive of $\csc^2 a x$


Then:

\(\displaystyle \int x \csc^2 a x \ \mathrm d x\) \(=\) \(\displaystyle \frac {-x \cot a x} a - \int \frac {-\cot a x} a \ \mathrm d x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {-x \cot a x} a + \frac 1 a \int \cot a x \ \mathrm d x + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {-x \cot a x} a + \frac 1 a \left({\frac 1 a \ln \left\vert{\sin a x}\right\vert}\right) + C\) Primitive of $\cot a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-x \cot a x} a + \frac 1 {a^2} \ln \left\vert{\sin a x}\right\vert + C\) simplifying

$\blacksquare$


Also see


Sources