integral of x\sqrt[4]{6x+7}

Lösung

\int x 4 6 x + 7 d x

\frac{1}{9} (\frac{1}{9} (6 x + 7)^{\frac{9}{4}} - \frac{7}{5} (6 x + 7)^{\frac{5}{4}}) + C

Schritte zur Lösung

\int x 4 6 x + 7 d x

Wende U-Substitution an

= \int \frac{u ^{4} ( u ^{4} - 7 )}{9} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int u^{4} (u^{4} - 7) d u

Multipliziere aus

u^{4} (u^{4} - 7) : u^{8} - 7 u^{4}

= \frac{1}{9} \cdot \int u^{8} - 7 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{1}{9} (\int u^{8} d u - \int 7 u^{4} d u)

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

\int 7 u^{4} d u = \frac{7 u ^{5}}{5}

= \frac{1}{9} (\frac{u ^{9}}{9} - \frac{7 u ^{5}}{5})

Setze in

u = 4 6 x + 7

ein

= \frac{1}{9} (\frac{( 4 6 x + 7 ) ^{9}}{9} - \frac{7 ( 4 6 x + 7 ) ^{5}}{5})

Vereinfache

\frac{1}{9} (\frac{( 4 6 x + 7 ) ^{9}}{9} - \frac{7 ( 4 6 x + 7 ) ^{5}}{5}) : \frac{1}{9} (\frac{1}{9} (6 x + 7)^{\frac{9}{4}} - \frac{7}{5} (6 x + 7)^{\frac{5}{4}})

= \frac{1}{9} (\frac{1}{9} (6 x + 7)^{\frac{9}{4}} - \frac{7}{5} (6 x + 7)^{\frac{5}{4}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} (\frac{1}{9} (6 x + 7)^{\frac{9}{4}} - \frac{7}{5} (6 x + 7)^{\frac{5}{4}}) + C

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