intégrale de-2 a 2 de-sqrt(5-x^2)

Solution

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

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

Décimale

- 7.53574 \dots

étapes des solutions

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

Extraire la constante:

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

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

Appliquer la subsitution trigonométrique

= - \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

Extraire la constante:

\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

Récrire en utilisant des identités trigonométriques

= - 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

Extraire la constante:

\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

Appliquer la règle de la somme:

\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})))

Simplifier

- 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}

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