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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

= 2 \cdot \int \frac{x ^{3}}{1 + x ^{2}} d x

Wende U-Substitution an

= 2 \cdot \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

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

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

\int 1 d u = u

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

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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