What is the integral from 1 to sqrt(3 of)arctan(1/x) ?

The integral from 1 to sqrt(3 of)arctan(1/x) is (2sqrt(3)ln(2)+2pi-sqrt(3)pi)/(4sqrt(3))

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integral from 1 to sqrt(3 of)arctan(1/x)

\int_{1}^{3} arctan (\frac{1}{x}) d x

\frac{2 3 ln ( 2 ) + 2 π - 3 π}{4 3}

Decimal

0.46807 \dots

Solution steps

\int_{1}^{3} arctan (\frac{1}{x}) d x

Apply Integration By Parts

= [x arctan (\frac{1}{x}) - \int - \frac{x}{1 + x ^{2}} d x]_{1}^{3}

\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})]_{1}^{3}

Simplify

[x arctan (\frac{1}{x}) - (- \frac{1}{2} ln 1 + x^{2})]_{1}^{3} : [\frac{1}{2} (2 x arctan (\frac{1}{x}) + ln 1 + x^{2})]_{1}^{3}

= [\frac{1}{2} (2 x arctan (\frac{1}{x}) + ln 1 + x^{2})]_{1}^{3}

Compute the boundaries:

\frac{2 3 ln ( 2 ) + 2 π - 3 π}{4 3}

= \frac{2 3 ln ( 2 ) + 2 π - 3 π}{4 3}

What is the integral from 1 to sqrt(3 of)arctan(1/x) ?
The integral from 1 to sqrt(3 of)arctan(1/x) is (2sqrt(3)ln(2)+2pi-sqrt(3)pi)/(4sqrt(3))