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

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

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integral from 1 to 5 of 3/(x^2+4)

\int_{1}^{5} \frac{3}{x ^{2} + 4} d x

\frac{3}{2} (arctan (\frac{5}{2}) - arctan (\frac{1}{2}))

Decimal Notation

1.08996 \dots

Solution steps

\int_{1}^{5} \frac{3}{x ^{2} + 4} d x

Take the constant out:

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

= 3 \cdot \int_{1}^{5} \frac{1}{x ^{2} + 4} d x

Apply Integral Substitution

= 3 \cdot \int_{\frac{1}{2}}^{\frac{5}{2}} \frac{1}{2 ( u ^{2} + 1 )} d u

Take the constant out:

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

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

Use the common integral:

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

= 3 \cdot \frac{1}{2} [arctan (u)]_{\frac{1}{2}}^{\frac{5}{2}}

Simplify

3 \cdot \frac{1}{2} [arctan (u)]_{\frac{1}{2}}^{\frac{5}{2}} : \frac{3}{2} [arctan (u)]_{\frac{1}{2}}^{\frac{5}{2}}

= \frac{3}{2} [arctan (u)]_{\frac{1}{2}}^{\frac{5}{2}}

Compute the boundaries:

arctan (\frac{5}{2}) - arctan (\frac{1}{2})

= \frac{3}{2} (arctan (\frac{5}{2}) - arctan (\frac{1}{2}))

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