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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{a ^{2} - a}{3 - a ^{2}} : \frac{a ^{2}}{3 - a ^{2}} - \frac{a}{3 - a ^{2}}

= \int \frac{a ^{2}}{3 - a ^{2}} - \frac{a}{3 - a ^{2}} d a

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

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

\int \frac{a}{3 - a ^{2}} d a = - 3 - a^{2}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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