What is the integral from 7 to 9 of arctan(7/x) ?

The integral from 7 to 9 of arctan(7/x) is 9arctan(7/9)+7/2 ln(130)-(7pi)/4-7/2 ln(98)

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integral from 7 to 9 of arctan(7/x)

\int_{7}^{9} arctan (\frac{7}{x}) d x

9 arctan (\frac{7}{9}) + \frac{7}{2} ln (130) - \frac{7 π}{4} - \frac{7}{2} ln (98)

Solution steps

\int_{7}^{9} arctan (\frac{7}{x}) d x

Apply Integration By Parts

: [x arctan (\frac{7}{x}) - \int - \frac{7 x}{49 + x ^{2}} d x]_{7}^{9}

= [x arctan (\frac{7}{x}) - \int - \frac{7 x}{49 + x ^{2}} d x]_{7}^{9}

\int - \frac{7 x}{49 + x ^{2}} d x = - \frac{7}{2} ln 49 + x^{2}

= [x arctan (\frac{7}{x}) - (- \frac{7}{2} ln 49 + x^{2})]_{7}^{9}

Simplify

= [x arctan (\frac{7}{x}) + \frac{7}{2} ln 49 + x^{2}]_{7}^{9}

Compute the boundaries:

9 arctan (\frac{7}{9}) + \frac{7}{2} ln (130) - 7 \cdot \frac{π}{4} - \frac{7}{2} ln (98)

= 9 arctan (\frac{7}{9}) + \frac{7}{2} ln (130) - 7 \cdot \frac{π}{4} - \frac{7}{2} ln (98)

Simplify

= 9 arctan (\frac{7}{9}) + \frac{7}{2} ln (130) - \frac{7 π}{4} - \frac{7}{2} ln (98)

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What is the integral from 7 to 9 of arctan(7/x) ?
The integral from 7 to 9 of arctan(7/x) is 9arctan(7/9)+7/2 ln(130)-(7pi)/4-7/2 ln(98)