Primitive of x by Arccotangent of x over a
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Theorem
- $\ds \int x \arccot \frac x a \rd x = \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C$
Proof
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:
\(\ds u\) | \(=\) | \(\ds \arccot \frac x a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {-a} {x^2 + a^2}\) | Derivative of $\arccot \dfrac x a$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {x^2} 2\) | Primitive of Power |
Then:
\(\ds \int x \arccot \frac x a \rd x\) | \(=\) | \(\ds \frac {x^2} 2 \arccot \frac x a - \int \frac {x^2} 2 \paren {\frac {-a} {x^2 + a^2} } \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arccot \frac x a + \frac a 2 \int \frac {x^2 \rd x} {x^2 + a^2} + C\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arccot \frac x a + \frac a 2 \paren {x - a \arctan {\frac x a} } + C\) | Primitive of $\dfrac {x^2} {x^2 + a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} 2 \arccot \frac x a + \frac a 2 \paren {x - a \paren {\frac \pi 2 - \arccot {\frac x a} } } + C\) | Sum of Arctangent and Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 - \frac {a^2 \pi} 2 + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2 + a^2} 2 \arccot \frac x a + \frac {a x} 2 + C\) | subsuming $\dfrac {a^2 \pi} 2$ into the arbitrary constant |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.489$