What is the integral from 0 to 1/2 of (1-x^2)^{1/2} ?

The integral from 0 to 1/2 of (1-x^2)^{1/2} is (2pi+3sqrt(3))/(24)

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integral from 0 to 1/2 of (1-x^2)^{1/2}

\int_{0}^{\frac{1}{2}} (1 - x^{2})^{\frac{1}{2}} d x

\frac{2 π + 3 3}{24}

Decimal

0.47830 \dots

Solution steps

\int_{0}^{\frac{1}{2}} (1 - x^{2})^{\frac{1}{2}} d x

Apply Trigonometric Substitution

= \int_{0}^{\frac{π}{6}} cos^{2} (u) d u

Rewrite using trig identities

= \int_{0}^{\frac{π}{6}} \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{6}} 1 + cos (2 u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{\frac{π}{6}} 1 d u + \int_{0}^{\frac{π}{6}} cos (2 u) d u)

\int_{0}^{\frac{π}{6}} 1 d u = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 u) d u = \frac{3}{4}

= \frac{1}{2} (\frac{π}{6} + \frac{3}{4})

Simplify

\frac{1}{2} (\frac{π}{6} + \frac{3}{4}) : \frac{2 π + 3 3}{24}

= \frac{2 π + 3 3}{24}

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What is the integral from 0 to 1/2 of (1-x^2)^{1/2} ?
The integral from 0 to 1/2 of (1-x^2)^{1/2} is (2pi+3sqrt(3))/(24)