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

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

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

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

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

Decimal

- 7.53574 \dots

Solution steps

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

Take the constant out:

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

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

Apply Trigonometric Substitution

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

Rewrite using trig identities

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

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

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

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

Simplify

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

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

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