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Beliebt Rechnen >

integral of (x^3)/(sqrt(5x^2+\sqrt{2))}

Lösung

\int \frac{x ^{3}}{5 x ^{2} + 2} d x

Lösung

\frac{1}{50} (\frac{2}{3} (5 x^{2} + 2)^{\frac{3}{2}} - 5 \frac{2}{5 x ^{2} + 2} x^{2} - 2 \frac{2}{5 x ^{2} + 2} - 2 5 x^{2} + 2) + C

Schritte zur Lösung

\int \frac{x ^{3}}{5 x ^{2} + 2} d x

Wende U-Substitution an

= \int \frac{u - 2}{50 u} d u

Entferne die Konstante:

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

= \frac{1}{50} \cdot \int \frac{u - 2}{u} d u

Multipliziere aus

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

= \frac{1}{50} \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}{50} (\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}{50} (\frac{2}{3} u^{\frac{3}{2}} - (\frac{2}{u} u + 2 u))

Setze in

u = 5 x^{2} + 2

ein

= \frac{1}{50} (\frac{2}{3} (5 x^{2} + 2)^{\frac{3}{2}} - (\frac{2}{5 x ^{2} + 2} (5 x^{2} + 2) + 2 5 x^{2} + 2))

Multipliziere aus

\frac{2}{3} (5 x^{2} + 2)^{\frac{3}{2}} - (\frac{2}{5 x ^{2} + 2} (5 x^{2} + 2) + 2 5 x^{2} + 2) : \frac{2}{3} (5 x^{2} + 2)^{\frac{3}{2}} - 5 \frac{2}{5 x ^{2} + 2} x^{2} - 2 \frac{2}{5 x ^{2} + 2} - 2 5 x^{2} + 2

= \frac{1}{50} (\frac{2}{3} (5 x^{2} + 2)^{\frac{3}{2}} - 5 \frac{2}{5 x ^{2} + 2} x^{2} - 2 \frac{2}{5 x ^{2} + 2} - 2 5 x^{2} + 2)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{50} (\frac{2}{3} (5 x^{2} + 2)^{\frac{3}{2}} - 5 \frac{2}{5 x ^{2} + 2} x^{2} - 2 \frac{2}{5 x ^{2} + 2} - 2 5 x^{2} + 2) + C

Graph

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