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Beliebt Rechnen >

integral from 0 to x of sqrt(a^2-t^2)

Lösung

\int_{0}^{x} a^{2} - t^{2} d t

Lösung

a^{2} \frac{1}{2} (arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

Schritte zur Lösung

\int_{0}^{x} a^{2} - t^{2} d t

Trigonometrische Substitution anwenden

= \int_{0}^{a r c s i n (\frac{x}{a})} a^{2} cos^{2} (u) d u

Entferne die Konstante:

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

= a^{2} \cdot \int_{0}^{a r c s i n (\frac{x}{a})} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= a^{2} \cdot \int_{0}^{a r c s i n (\frac{x}{a})} \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

= a^{2} \frac{1}{2} \cdot \int_{0}^{a r c s i n (\frac{x}{a})} 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

= a^{2} \frac{1}{2} (\int_{0}^{a r c s i n (\frac{x}{a})} 1 d u + \int_{0}^{a r c s i n (\frac{x}{a})} cos (2 u) d u)

\int_{0}^{a r c s i n (\frac{x}{a})} 1 d u = arcsin (\frac{x}{a})

\int_{0}^{a r c s i n (\frac{x}{a})} cos (2 u) d u = \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

= a^{2} \frac{1}{2} (arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

Beliebte Beispiele

derivative of sqrt(x^2-25)

\frac{d}{d x} (x^{2} - 25)

tangent sin(7x)+cos(6x),\at pi/6

t an g e n t sin (7 x) + cos (6 x), at \frac{π}{6}

derivative y=x^2sqrt(8+x)

d er i v a t i v e y = x^{2} 8 + x

sum from n=1 to infinity of ln(1/(n+1))

n = 1 \sum \infty ln (\frac{1}{n + 1})

integral of cosh(x/(27.2632))

\int cosh (\frac{x}{27.2632}) d x