What is the integral from 1 to 3 of 3/(5x^2+5) ?

The integral from 1 to 3 of 3/(5x^2+5) is 3/5 (arctan(3)-pi/4)

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integral from 1 to 3 of 3/(5x^2+5)

\int_{1}^{3} \frac{3}{5 x ^{2} + 5} d x

\frac{3}{5} (arctan (3) - \frac{π}{4})

Solution steps

\int_{1}^{3} \frac{3}{5 x ^{2} + 5} d x

Take the constant out:

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

= 3 \cdot \int_{1}^{3} \frac{1}{5 x ^{2} + 5} d x

\frac{1}{5 x ^{2} + 5} = \frac{1}{5} \frac{1}{x ^{2} + 1}

= 3 \cdot \int_{1}^{3} \frac{1}{5} \cdot \frac{1}{x ^{2} + 1} d x

Take the constant out:

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

= 3 \cdot \frac{1}{5} \cdot \int_{1}^{3} \frac{1}{x ^{2} + 1} d x

Use the common integral:

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

= 3 \cdot \frac{1}{5} [arctan (x)]_{1}^{3}

Simplify

3 \cdot \frac{1}{5} [arctan (x)]_{1}^{3} : \frac{3}{5} [arctan (x)]_{1}^{3}

= \frac{3}{5} [arctan (x)]_{1}^{3}

Compute the boundaries:

arctan (3) - \frac{π}{4}

= \frac{3}{5} (arctan (3) - \frac{π}{4})

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What is the integral from 1 to 3 of 3/(5x^2+5) ?
The integral from 1 to 3 of 3/(5x^2+5) is 3/5 (arctan(3)-pi/4)