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

Lösung

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

\frac{π}{4} - \frac{1}{2}

Dezimale

0.28539 \dots

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} \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

= 2 \cdot \frac{1}{2} \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} 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

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

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

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

= 2 \cdot \frac{1}{2} (\frac{π}{4} - \frac{1}{2})

Vereinfache

2 \cdot \frac{1}{2} (\frac{π}{4} - \frac{1}{2}) : \frac{π}{4} - \frac{1}{2}

= \frac{π}{4} - \frac{1}{2}

\int_{1}^{3} \frac{x}{( x ^{2} + 4 ) ^{3}} d x

\int_{- π}^{0} sin (3 x) cos (2 x) d x

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

\int_{1}^{x} \frac{3}{t ^{3} + 1} d t

\int_{0}^{\frac{π}{2}} 5 cos^{2} (θ) d θ