What is the integral from 1 to 3 of 1/(x^2+81) ?

The integral from 1 to 3 of 1/(x^2+81) is 1/9 (arctan(1/3)-arctan(1/9))

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integral from 1 to 3 of 1/(x^2+81)

\int_{1}^{3} \frac{1}{x ^{2} + 81} d x

\frac{1}{9} (arctan (\frac{1}{3}) - arctan (\frac{1}{9}))

Decimal Notation

0.02345 \dots

Solution steps

\int_{1}^{3} \frac{1}{x ^{2} + 81} d x

Apply Integral Substitution

= \int_{\frac{1}{9}}^{\frac{1}{3}} \frac{1}{9 ( u ^{2} + 1 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{9} \cdot \int_{\frac{1}{9}}^{\frac{1}{3}} \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{1}{9} [arctan (u)]_{\frac{1}{9}}^{\frac{1}{3}}

Compute the boundaries:

arctan (\frac{1}{3}) - arctan (\frac{1}{9})

= \frac{1}{9} (arctan (\frac{1}{3}) - arctan (\frac{1}{9}))

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What is the integral from 1 to 3 of 1/(x^2+81) ?
The integral from 1 to 3 of 1/(x^2+81) is 1/9 (arctan(1/3)-arctan(1/9))