What is the integral from 4 to 6 of arctan(4/x) ?

The integral from 4 to 6 of arctan(4/x) is 6arctan(2/3)+ln(169/64)-pi

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integral from 4 to 6 of arctan(4/x)

\int_{4}^{6} arctan (\frac{4}{x}) d x

6 arctan (\frac{2}{3}) + ln (\frac{169}{64}) - π

Decimal Notation

1.35743 \dots

Solution steps

\int_{4}^{6} arctan (\frac{4}{x}) d x

Apply Integration By Parts

: [x arctan (\frac{4}{x}) - \int - \frac{4 x}{16 + x ^{2}} d x]_{4}^{6}

= [x arctan (\frac{4}{x}) - \int - \frac{4 x}{16 + x ^{2}} d x]_{4}^{6}

\int - \frac{4 x}{16 + x ^{2}} d x = - 2 ln 16 + x^{2}

= [x arctan (\frac{4}{x}) - (- 2 ln 16 + x^{2})]_{4}^{6}

Simplify

= [x arctan (\frac{4}{x}) + 2 ln 16 + x^{2}]_{4}^{6}

Compute the boundaries:

6 arctan (\frac{2}{3}) + 2 ln (52) - π - 10 ln (2)

= 6 arctan (\frac{2}{3}) + 2 ln (52) - π - 10 ln (2)

Simplify

= 6 arctan (\frac{2}{3}) + ln (\frac{169}{64}) - π

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What is the integral from 4 to 6 of arctan(4/x) ?
The integral from 4 to 6 of arctan(4/x) is 6arctan(2/3)+ln(169/64)-pi