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Beliebt Rechnen >

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

Lösung

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

Lösung

2 (\frac{1}{7} (x + 1)^{\frac{7}{2}} - \frac{3}{5} (x + 1)^{\frac{5}{2}} + (x + 1)^{\frac{3}{2}} - x + 1) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= 2 \cdot \int (u^{2} - 1)^{3} d u

Multipliziere aus

(u^{2} - 1)^{3} : u^{6} - 3 u^{4} + 3 u^{2} - 1

= 2 \cdot \int u^{6} - 3 u^{4} + 3 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^{6} d u - \int 3 u^{4} d u + \int 3 u^{2} d u - \int 1 d u)

\int u^{6} d u = \frac{u ^{7}}{7}

\int 3 u^{4} d u = \frac{3 u ^{5}}{5}

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

\int 1 d u = u

= 2 (\frac{u ^{7}}{7} - \frac{3 u ^{5}}{5} + u^{3} - u)

Setze in

u = x + 1

ein

= 2 (\frac{( x + 1 ) ^{7}}{7} - \frac{3 ( x + 1 ) ^{5}}{5} + (x + 1)^{3} - x + 1)

Vereinfache

2 (\frac{( x + 1 ) ^{7}}{7} - \frac{3 ( x + 1 ) ^{5}}{5} + (x + 1)^{3} - x + 1) : 2 (\frac{1}{7} (x + 1)^{\frac{7}{2}} - \frac{3}{5} (x + 1)^{\frac{5}{2}} + (x + 1)^{\frac{3}{2}} - x + 1)

= 2 (\frac{1}{7} (x + 1)^{\frac{7}{2}} - \frac{3}{5} (x + 1)^{\frac{5}{2}} + (x + 1)^{\frac{3}{2}} - x + 1)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{7} (x + 1)^{\frac{7}{2}} - \frac{3}{5} (x + 1)^{\frac{5}{2}} + (x + 1)^{\frac{3}{2}} - x + 1) + C

Graph

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