integral of ((x^2))/((a^2-x^2)^{3/2)}

Lösung

\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})) + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

\int \frac{10 s + 10}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

in t e g r a l x - 1

\int \frac{x ^{2} - x ^{5} + x}{x ^{5}} d x

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