integral from-1 to 1 of 2sqrt(1-(x^2)/2)

Lösung

\int_{- 1}^{1} 2 1 - \frac{x ^{2}}{2} d x

2 (\frac{π}{2} + 1)

Dezimale

3.63565 \dots

Schritte zur Lösung

\int_{- 1}^{1} 2 1 - \frac{x ^{2}}{2} d x

Entferne die Konstante:

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

= 2 \cdot \int_{- 1}^{1} 1 - \frac{x ^{2}}{2} d x

Trigonometrische Substitution anwenden

= 2 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} 2 cos^{2} (u) d u

Entferne die Konstante:

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

= 22 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 22 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{1 + cos ( 2 u )}{2} d u

Entferne die Konstante:

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

= 22 \frac{1}{2} \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} 1 + cos (2 u) d u

Wende die Summenregel an:

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

= 22 \frac{1}{2} (\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d u + \int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 u) d u)

\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d u = \frac{π}{2}

\int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 u) d u = 1

= 22 \frac{1}{2} (\frac{π}{2} + 1)

Vereinfache

22 \frac{1}{2} (\frac{π}{2} + 1) : 2 (\frac{π}{2} + 1)

= 2 (\frac{π}{2} + 1)

t an g e n t \frac{12 x}{x ^{6}}

x \to 1 lim (ln (x^{2}))

f (x) = sin^{4} (3 x + 5)

x \to 0 lim (3 x csc (5 x))

\frac{\partial}{\partial x} (\frac{c}{x})