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Beliebt Rechnen >

integral of (5x^2)/((a^2-x^2)^{3/2)}

Lösung

\int \frac{5 x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} d x

Lösung

5 (\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{5 x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} d x

Wende die partielle Integration an

= 5 (\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}))

= 5 (\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

= 5 (\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

= 5 (\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

Graph

Beliebte Beispiele

tangent f(x)=e^x,\at x=ln(10)

t an g e n t f (x) = e^{x}, at x = ln (10)

derivative tan(sin(x^3))

d er i v a t i v e tan (sin (x^{3}))

(dy}{dx}=-\frac{y+1)/x

\frac{d y}{d x} = - \frac{y + 1}{x}

derivative f(x)=(x^2+1)^3(x^2+2)^6

d er i v a t i v e f (x) = (x^{2} + 1)^{3} (x^{2} + 2)^{6}

derivative of xlog_{10}(2/x)

\frac{d}{d x} (x lo g_{10} (\frac{2}{x}))