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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

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

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

Multipliziere aus

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

Wende die Summenregel an:

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of sec^2(x)(1+tan(x))

\int sec^{2} (x) (1 + tan (x)) d x

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

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

integral 1/(x^2)

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

integral of (1-x)/(sqrt(9-4x^2))

\int \frac{1 - x}{9 - 4 x ^{2}} d x

integral of x/(sqrt(2-3x^2))

\int \frac{x}{2 - 3 x ^{2}} d x