What is the integral of arctan(1/x) ?

The integral of arctan(1/x) is xarctan(1/x)+1/2 ln|1+x^2|+C

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integral of arctan(1/x)

integral

arctan (\frac{1}{x})

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

Solution steps

\int arctan (\frac{1}{x}) d x

Apply Integration By Parts

: x arctan (\frac{1}{x}) - \int - \frac{x}{1 + x ^{2}} d x

= x arctan (\frac{1}{x}) - \int - \frac{x}{1 + x ^{2}} d x

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

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

Simplify

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

Add a constant to the solution

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

\frac{x ^{2}}{( 1 + x )}

\frac{( x + 4 )}{( x ^{2} + 1 )}

sin^{2} (4 x)

\frac{- 2}{x ^{2}}

\frac{2}{e ^{x}}

What is the integral of arctan(1/x) ?
The integral of arctan(1/x) is xarctan(1/x)+1/2 ln|1+x^2|+C