What is the integral from 0 to 4 of 1/(9x^2+25) ?

The integral from 0 to 4 of 1/(9x^2+25) is 1/15 arctan(12/5)

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integral from 0 to 4 of 1/(9x^2+25)

\int_{0}^{4} \frac{1}{9 x ^{2} + 25} d x

\frac{1}{15} arctan (\frac{12}{5})

Decimal

0.07840 \dots

Solution steps

\int_{0}^{4} \frac{1}{9 x ^{2} + 25} d x

Apply Integral Substitution

= \int_{0}^{\frac{12}{5}} \frac{1}{15 ( 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}{15} \cdot \int_{0}^{\frac{12}{5}} \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{15} [arctan (u)]_{0}^{\frac{12}{5}}

Compute the boundaries:

arctan (\frac{12}{5})

= \frac{1}{15} arctan (\frac{12}{5})

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What is the integral from 0 to 4 of 1/(9x^2+25) ?
The integral from 0 to 4 of 1/(9x^2+25) is 1/15 arctan(12/5)