What is the integral of arccot(x/n) ?

The integral of arccot(x/n) is xarccot(x/n)+1/2 nln|(x^2)/(n^2)+1|+C

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integral of arccot(x/n)

\int arccot (\frac{x}{n}) d x

x arccot (\frac{x}{n}) + \frac{1}{2} n ln \frac{x ^{2}}{n ^{2}} + 1 + C

Solution steps

\int arccot (\frac{x}{n}) d x

Apply u-substitution

: \int arccot (u) n d u

= \int arccot (u) n d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= n \cdot \int arccot (u) d u

Apply Integration By Parts

: u arccot (u) - \int - \frac{u}{u ^{2} + 1} d u

= n (u arccot (u) - \int - \frac{u}{u ^{2} + 1} d u)

\int - \frac{u}{u ^{2} + 1} d u = - \frac{1}{2} ln u^{2} + 1

= n (u arccot (u) - (- \frac{1}{2} ln u^{2} + 1))

Substitute back

u = \frac{x}{n}

= n (\frac{x}{n} arccot (\frac{x}{n}) - (- \frac{1}{2} ln (\frac{x}{n})^{2} + 1))

Simplify

n (\frac{x}{n} arccot (\frac{x}{n}) - (- \frac{1}{2} ln (\frac{x}{n})^{2} + 1)) : x arccot (\frac{x}{n}) + \frac{1}{2} n ln \frac{x ^{2}}{n ^{2}} + 1

= x arccot (\frac{x}{n}) + \frac{1}{2} n ln \frac{x ^{2}}{n ^{2}} + 1

Add a constant to the solution

= x arccot (\frac{x}{n}) + \frac{1}{2} n ln \frac{x ^{2}}{n ^{2}} + 1 + C

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What is the integral of arccot(x/n) ?
The integral of arccot(x/n) is xarccot(x/n)+1/2 nln|(x^2)/(n^2)+1|+C