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Beliebt Rechnen >

integral of (x^2)/(1+sqrt(1-x^2))

Lösung

\int \frac{x ^{2}}{1 + 1 - x ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2}}{1 + 1 - x ^{2}} d x

Multipliziere mit der Konjugation von

1 + 1 - x^{2} : \frac{x ^{2}}{1 - 1 - x ^{2}}

= \int \frac{x ^{2}}{\frac{x ^{2}}{1 - 1 - x ^{2}}} d x

Vereinfache

\frac{x ^{2}}{\frac{x ^{2}}{1 - 1 - x ^{2}}} : 1 - - x^{2} + 1

= \int 1 - - x^{2} + 1 d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x - \int - x^{2} + 1 d x

\int 1 d x = x

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of ((-x-1))/(x^2+3)

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

integral of xsqrt(8-2x^2)

\int x 8 - 2 x^{2} d x

integral of (-2x+11)/(x^2-8x+25)

\int \frac{- 2 x + 11}{x ^{2} - 8 x + 25} d x

integral of 1-2x^2

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

integral of (1/(x+2))

\int (\frac{1}{x + 2}) d x