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Beliebt Rechnen >

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

Lösung

\int x 4 3 x + 5 d x

Lösung

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

Schritte zur Lösung

\int x 4 3 x + 5 d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= \frac{4}{9} \cdot \int u^{4} (u^{4} - 5) d u

Multipliziere aus

u^{4} (u^{4} - 5) : u^{8} - 5 u^{4}

= \frac{4}{9} \cdot \int u^{8} - 5 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}{9} (\int u^{8} d u - \int 5 u^{4} d u)

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

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

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

Setze in

u = 4 3 x + 5

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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