Primitive of Arccotangent Function
Theorem
- $\ds \int \arccot x \rd x = x \arccot x + \frac {\map \ln {x^2 + 1} } 2 + C$
Corollary
- $\ds \int \arccot \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$
Proof 1
Let:
| \(\ds u\) | \(=\) | \(\ds \arccot x\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \cot u\) | \(=\) | \(\ds x\) | Definition of Arccotangent | |||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \csc u\) | \(=\) | \(\ds \sqrt {1 + x^2}\) | Difference of Squares of Cosecant and Cotangent |
Then:
| \(\ds \int \arccot x \rd x\) | \(=\) | \(\ds -\int u \csc^2 u \rd u\) | Primitive of Function of Arccotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {-u \cot u + \ln \size {\sin u} } + C\) | Primitive of $x \csc^2 a x$ with $a := 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds u \cot u - \ln \size {\sin u} + C\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds u \cot u + \ln \size {\csc u} + C\) | Logarithm of Reciprocal and Cosecant is Reciprocal of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds u x + a \ln \size {\csc u} + C\) | Substitution for $\cot u$ from $(1)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds x u + a \ln \size {\sqrt {1 + x^2} } + C\) | Substitution for $\csc u$ from $(2)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccot x + \ln \size {\sqrt {1 + x^2} } + C\) | Substitution for $u$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccot x + \frac 1 2 \ln \size {x^2 + 1} + C\) | Logarithm of Power and simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccot x + \frac {\map \ln {x^2 + 1} } 2 + C\) | $x^2 + 1$ always positive |
$\blacksquare$
Proof 2
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
| \(\ds u\) | \(=\) | \(\ds \arccot x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {-1} {x^2 + 1}\) | Derivative of $\arccot \dfrac x a$, setting $a := 1$ |
and let:
| \(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds x\) | Primitive of Constant |
Then:
| \(\ds \int \arccot x \rd x\) | \(=\) | \(\ds x \arccot x - \int x \paren {\frac {-1} {x^2 + 1} } \rd x + C\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccot x + \int \frac {x \rd x} {x^2 + 1} + C\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccot x + \paren {\frac 1 2 \map \ln {x^2 + 1} } + C\) | Primitive of $\dfrac x {x^2 + a^2}$, setting $a := 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \arccot x + \frac a 2 \map \ln {x^2 + 1} + C\) | simplifying |
$\blacksquare$
Also presented as
This result can also be presented as:
- $\ds \int \arccot x \rd x = x \arccot x + \ln \sqrt {x^2 + 1} + C$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals