Primitive of Function of Arccosecant
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Theorem
- $\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$
where $u = \arccsc \dfrac x a$.
Proof
First note that:
| \(\ds u\) | \(=\) | \(\ds \arccsc \frac x a\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a \csc u\) | Definition of Arccosecant |
Then:
| \(\ds u\) | \(=\) | \(\ds \arccsc \frac x a\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {-a} {\size x {\sqrt {x^2 - a^2} } }\) | Derivative of Arccosecant Function: Corollary | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \map F {\arccsc \frac x a} \rd x\) | \(=\) | \(\ds \int \map F u \frac {\size x {\sqrt {x^2 - a^2} } } {-a} \rd u\) | Primitive of Composite Function | ||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u \frac {\size {a \csc u} {\sqrt {a^2 \csc^2 u - a^2} } } {-a} \rd u\) | Definition of $x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u \size {\csc u} \paren {-\sqrt {a^2 \csc^2 u - a^2} } \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u \paren {-a} \size {\csc u} {\sqrt {\csc^2 u - 1} } \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \map F u \paren {-a} \size {\csc u} \cot u \rd u\) | Difference of Squares of Cosecant and Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -a \int \map F u \size {\csc u} \cot u \rd u\) | Primitive of Constant Multiple of Function |
$\blacksquare$