What is the integral from 0 to 3/2 pi of 2/(9+x^2) ?

The integral from 0 to 3/2 pi of 2/(9+x^2) is 2/3 arctan(pi/2)

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integral from 0 to 3/2 pi of 2/(9+x^2)

\int_{0}^{\frac{3}{2} π} \frac{2}{9 + x ^{2}} d x

\frac{2}{3} arctan (\frac{π}{2})

Decimal

0.66925 \dots

Solution steps

\int_{0}^{\frac{3}{2} π} \frac{2}{9 + x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{\frac{3}{2} π} \frac{1}{9 + x ^{2}} d x

Apply Integral Substitution

= 2 \cdot \int_{0}^{\frac{π}{2}} \frac{1}{3 ( u ^{2} + 1 )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int_{0}^{\frac{π}{2}} \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= 2 \cdot \frac{1}{3} [arctan (u)]_{0}^{\frac{π}{2}}

Simplify

2 \cdot \frac{1}{3} [arctan (u)]_{0}^{\frac{π}{2}} : \frac{2}{3} [arctan (u)]_{0}^{\frac{π}{2}}

= \frac{2}{3} [arctan (u)]_{0}^{\frac{π}{2}}

Compute the boundaries:

arctan (\frac{π}{2})

= \frac{2}{3} arctan (\frac{π}{2})

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What is the integral from 0 to 3/2 pi of 2/(9+x^2) ?
The integral from 0 to 3/2 pi of 2/(9+x^2) is 2/3 arctan(pi/2)