What is the integral from 2 to 4 of arctan(2/x) ?

The integral from 2 to 4 of arctan(2/x) is 4arctan(1/2)+ln(5/2)-pi/2

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integral from 2 to 4 of arctan(2/x)

\int_{2}^{4} arctan (\frac{2}{x}) d x

4 arctan (\frac{1}{2}) + ln (\frac{5}{2}) - \frac{π}{2}

Decimal Notation

1.20008 \dots

Solution steps

\int_{2}^{4} arctan (\frac{2}{x}) d x

Apply Integration By Parts

: [x arctan (\frac{2}{x}) - \int - \frac{2 x}{4 + x ^{2}} d x]_{2}^{4}

= [x arctan (\frac{2}{x}) - \int - \frac{2 x}{4 + x ^{2}} d x]_{2}^{4}

\int - \frac{2 x}{4 + x ^{2}} d x = - ln 4 + x^{2}

= [x arctan (\frac{2}{x}) - (- ln 4 + x^{2})]_{2}^{4}

Simplify

= [x arctan (\frac{2}{x}) + ln 4 + x^{2}]_{2}^{4}

Compute the boundaries:

4 arctan (\frac{1}{2}) + ln (20) - \frac{π}{2} - 3 ln (2)

= 4 arctan (\frac{1}{2}) + ln (20) - \frac{π}{2} - 3 ln (2)

Simplify

= 4 arctan (\frac{1}{2}) + ln (\frac{5}{2}) - \frac{π}{2}

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What is the integral from 2 to 4 of arctan(2/x) ?
The integral from 2 to 4 of arctan(2/x) is 4arctan(1/2)+ln(5/2)-pi/2