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

Lösung

integral

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

\frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - \int \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} d x

\int \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} d x = \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

= \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - (\frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

Vereinfache

= \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

Füge eine Konstante zur Lösung hinzu

= \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})) + C

in t e g r a l \frac{x}{( 1 + x ^{2} ) ^{2}}

in t e g r a l \frac{x ^{2}}{( 1 + x )}

in t e g r a l \frac{- 2}{x ^{2}}

in t e g r a l \frac{2}{e ^{x}}

in t e g r a l cos (\frac{1}{x})