integral of x^5\sqrt[4]{1+x^3}

Lösung

\int x^{5} 4 1 + x^{3} d x

\frac{4}{3} (\frac{1}{9} (1 + x^{3})^{\frac{9}{4}} - \frac{1}{5} (1 + x^{3})^{\frac{5}{4}}) + C

Schritte zur Lösung

\int x^{5} 4 1 + x^{3} d x

Wende U-Substitution an

= \int \frac{4 u ^{4} ( u ^{4} - 1 )}{3} d u

Entferne die Konstante:

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

= \frac{4}{3} \cdot \int u^{4} (u^{4} - 1) d u

Multipliziere aus

u^{4} (u^{4} - 1) : u^{8} - u^{4}

= \frac{4}{3} \cdot \int u^{8} - u^{4} 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{4}{3} (\int u^{8} d u - \int u^{4} d u)

\int u^{8} d u = \frac{u ^{9}}{9}

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

= \frac{4}{3} (\frac{u ^{9}}{9} - \frac{u ^{5}}{5})

Setze in

u = 4 1 + x^{3}

ein

= \frac{4}{3} (\frac{( 4 1 + x ^{3} ) ^{9}}{9} - \frac{( 4 1 + x ^{3} ) ^{5}}{5})

Vereinfache

\frac{4}{3} (\frac{( 4 1 + x ^{3} ) ^{9}}{9} - \frac{( 4 1 + x ^{3} ) ^{5}}{5}) : \frac{4}{3} (\frac{1}{9} (1 + x^{3})^{\frac{9}{4}} - \frac{1}{5} (1 + x^{3})^{\frac{5}{4}})

= \frac{4}{3} (\frac{1}{9} (1 + x^{3})^{\frac{9}{4}} - \frac{1}{5} (1 + x^{3})^{\frac{5}{4}})

Füge eine Konstante zur Lösung hinzu

= \frac{4}{3} (\frac{1}{9} (1 + x^{3})^{\frac{9}{4}} - \frac{1}{5} (1 + x^{3})^{\frac{5}{4}}) + C

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