integral of x\sqrt[5]{(x+1)^2}

Lösung

\int x 5 (x + 1)^{2} d x

\frac{5}{12} (x + 1)^{\frac{12}{5}} - \frac{5}{7} (x + 1)^{\frac{7}{5}} + C

Schritte zur Lösung

\int x 5 (x + 1)^{2} d x

5 (x + 1)^{2} = (x + 1)^{\frac{2}{5}},

angenommen

(x + 1) \geq 0

= \int x (x + 1)^{\frac{2}{5}} d x

Wende U-Substitution an

= \int (u - 1) u^{\frac{2}{5}} d u

Multipliziere aus

(u - 1) u^{\frac{2}{5}} : u^{\frac{7}{5}} - u^{\frac{2}{5}}

= \int u^{\frac{7}{5}} - u^{\frac{2}{5}} 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

= \int u^{\frac{7}{5}} d u - \int u^{\frac{2}{5}} d u

\int u^{\frac{7}{5}} d u = \frac{5}{12} u^{\frac{12}{5}}

\int u^{\frac{2}{5}} d u = \frac{5}{7} u^{\frac{7}{5}}

= \frac{5}{12} u^{\frac{12}{5}} - \frac{5}{7} u^{\frac{7}{5}}

Setze in

u = x + 1

ein

= \frac{5}{12} (x + 1)^{\frac{12}{5}} - \frac{5}{7} (x + 1)^{\frac{7}{5}}

Füge eine Konstante zur Lösung hinzu

= \frac{5}{12} (x + 1)^{\frac{12}{5}} - \frac{5}{7} (x + 1)^{\frac{7}{5}} + C

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