integral of (x^3)/(sqrt(x^2-1))

Lösung

\int \frac{x ^{3}}{x ^{2} - 1} d x

\frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1 + C

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} - 1} d x

Wende U-Substitution an

= \int u^{2} + 1 d u

Wende die Summenregel an:

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

= \int u^{2} d u + \int 1 d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

= \frac{u ^{3}}{3} + u

Setze in

u = x^{2} - 1

ein

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

Vereinfache

\frac{( x ^{2} - 1 ) ^{3}}{3} + x^{2} - 1 : \frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1

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

Füge eine Konstante zur Lösung hinzu

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

p a r i t y 25 - x^{2} d x

\frac{\partial}{\partial y} (x e^{2 yz})

\frac{\partial}{\partial s} (\frac{s}{s ^{2} + t ^{2}})

\frac{d y}{d x} = \frac{( x ^{2} )}{y ^{2}}

\frac{\partial}{\partial z} (z^{2})