What is the integral from 0 to 1 of arctan(1/x) ?

The integral from 0 to 1 of arctan(1/x) is pi/4+1/2 ln(2)

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integral from 0 to 1 of arctan(1/x)

\int_{0}^{1} arctan (\frac{1}{x}) d x

\frac{π}{4} + \frac{1}{2} ln (2)

Decimal

1.13197 \dots

Solution steps

\int_{0}^{1} arctan (\frac{1}{x}) d x

Apply Integration By Parts

= [x arctan (\frac{1}{x}) - \int - \frac{x}{1 + x ^{2}} d x]_{0}^{1}

\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})]_{0}^{1}

Simplify

= [x arctan (\frac{1}{x}) + \frac{1}{2} ln 1 + x^{2}]_{0}^{1}

Compute the boundaries:

\frac{π}{4} + \frac{1}{2} ln (2)

= \frac{π}{4} + \frac{1}{2} ln (2)

What is the integral from 0 to 1 of arctan(1/x) ?
The integral from 0 to 1 of arctan(1/x) is pi/4+1/2 ln(2)