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Beliebt Rechnen >

integral of (x^3)/(sqrt(x^2+\sqrt{2))}

Lösung

\int \frac{x ^{3}}{x ^{2} + 2} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} + 2} d x

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int \frac{2}{u} d u = \frac{2}{u} u + 2 u

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

Setze in

u = x^{2} + 2

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (cos(ax))/(1-sin(ax))

\int \frac{cos ( a x )}{1 - sin ( a x )} d x

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