integral (a-x^2)^{1/2}

Lösung

integral

(a - x^{2})^{\frac{1}{2}}

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

Schritte zur Lösung

\int (a - x^{2})^{\frac{1}{2}} d x

Trigonometrische Substitution anwenden

= \int a cos^{2} (u) d u

Entferne die Konstante:

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

= a \cdot \int cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= a \cdot \int \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 \frac{1}{2} \cdot \int 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 \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= a \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Setze in

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

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l sin^{3} (x) cos^{3} (x)

f (x) = (\frac{( x - 1 )}{( x + 1 )})^{\frac{1}{2}}

in t e g r a l sin^{2} (x) cos^{3} (x)

in t e g r a l \frac{1}{v ^{4}}

in t e g r a l \frac{( x ^{3} + 2 )}{( 4 - x ^{2} )}