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