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

Lösung

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

\frac{3}{2} arcsin (x) - \frac{1}{4} sin (2 arcsin (x)) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

arcsin (x) + \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) : \frac{3}{2} arcsin (x) - \frac{1}{4} sin (2 arcsin (x))

= \frac{3}{2} arcsin (x) - \frac{1}{4} sin (2 arcsin (x))

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} arcsin (x) - \frac{1}{4} sin (2 arcsin (x)) + C

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