integral (x^3)/((1+x^2)^{1/2)}

Lösung

integral

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u - 1}{2 u ^{\frac{1}{2}}} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u - 1}{u ^{\frac{1}{2}}} d u

Multipliziere aus

\frac{u - 1}{u ^{\frac{1}{2}}} : u^{\frac{1}{2}} - \frac{1}{u ^{\frac{1}{2}}}

= \frac{1}{2} \cdot \int u^{\frac{1}{2}} - \frac{1}{u ^{\frac{1}{2}}} 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

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

\int u^{\frac{1}{2}} d u = \frac{2}{3} u^{\frac{3}{2}}

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

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

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{5}{( 3 + 4 x ^{2} )}

in t e g r a l \frac{1}{( x ^{\frac{1}{3}} + 1 )}

in t e g r a l 2 x cos (x^{2} - 5)

in t e g r a l arctan (x^{2})

in t e g r a l \frac{( e ^{- x} ) ^{2}}{b}