What is the integral from 0 to 2 of sqrt(5-x^2) ?

The integral from 0 to 2 of sqrt(5-x^2) is (5(2arcsin(2/(sqrt(5)))+sin(2arcsin(2/(sqrt(5))))))/4

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integral from 0 to 2 of sqrt(5-x^2)

\int_{0}^{2} 5 - x^{2} d x

\frac{5 ( 2 arcsin ( \frac{2}{5} ) + sin ( 2 arcsin ( \frac{2}{5} ) ) )}{4}

Decimal Notation

3.76787 \dots

Solution steps

\int_{0}^{2} 5 - x^{2} d x

Apply Trigonometric Substitution

: \int_{0}^{a r c s i n (\frac{2}{5})} 5 cos^{2} (u) d u

= \int_{0}^{a r c s i n (\frac{2}{5})} 5 cos^{2} (u) d u

Take the constant out:

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

= 5 \cdot \int_{0}^{a r c s i n (\frac{2}{5})} cos^{2} (u) d u

Rewrite using trig identities

= 5 \cdot \int_{0}^{a r c s i n (\frac{2}{5})} \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

= 5 \cdot \frac{1}{2} \cdot \int_{0}^{a r c s i n (\frac{2}{5})} 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

= 5 \cdot \frac{1}{2} (\int_{0}^{a r c s i n (\frac{2}{5})} 1 d u + \int_{0}^{a r c s i n (\frac{2}{5})} cos (2 u) d u)

\int_{0}^{a r c s i n (\frac{2}{5})} 1 d u = arcsin (\frac{2}{5})

\int_{0}^{a r c s i n (\frac{2}{5})} cos (2 u) d u = \frac{sin ( 2 arcsin ( \frac{2}{5} ) )}{2}

= 5 \cdot \frac{1}{2} arcsin (\frac{2}{5}) + \frac{sin ( 2 arcsin ( \frac{2}{5} ) )}{2}

Simplify

5 \cdot \frac{1}{2} arcsin (\frac{2}{5}) + \frac{sin ( 2 arcsin ( \frac{2}{5} ) )}{2} : \frac{5 ( 2 arcsin ( \frac{2}{5} ) + sin ( 2 arcsin ( \frac{2}{5} ) ) )}{4}

= \frac{5 ( 2 arcsin ( \frac{2}{5} ) + sin ( 2 arcsin ( \frac{2}{5} ) ) )}{4}

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What is the integral from 0 to 2 of sqrt(5-x^2) ?
The integral from 0 to 2 of sqrt(5-x^2) is (5(2arcsin(2/(sqrt(5)))+sin(2arcsin(2/(sqrt(5))))))/4