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

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

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

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

\frac{5 π}{3 3}

Decimal

3.02299 \dots

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

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

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

Compute the boundaries:

\frac{π}{3}

= \frac{5}{3} \cdot \frac{π}{3}

Simplify

= \frac{5 π}{3 3}

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