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

Lösung

\int_{c o s (x)}^{1} 1 - t^{2} d t

\frac{1}{2} (\frac{π}{2} - arcsin (cos (x)) - \frac{1}{2} sin (2 arcsin (cos (x))))

Schritte zur Lösung

\int_{c o s (x)}^{1} 1 - t^{2} d t

Trigonometrische Substitution anwenden

= \int_{a r c s i n (c o s (x))}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{a r c s i n (c o s (x))}^{\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

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

= \frac{1}{2} (\int_{a r c s i n (c o s (x))}^{\frac{π}{2}} 1 d u + \int_{a r c s i n (c o s (x))}^{\frac{π}{2}} cos (2 u) d u)

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

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

= \frac{1}{2} (\frac{π}{2} - arcsin (cos (x)) - \frac{1}{2} sin (2 arcsin (cos (x))))

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