integral of t\sqrt[3]{2t+7}

Lösung

\int t 3 2 t + 7 d t

\frac{1}{4} (\frac{3}{7} (2 t + 7)^{\frac{7}{3}} - \frac{21}{4} (2 t + 7)^{\frac{4}{3}}) + C

Schritte zur Lösung

\int t 3 2 t + 7 d t

Wende U-Substitution an

= \int \frac{3 u ( u - 7 )}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int 3 u (u - 7) d u

Multipliziere aus

3 u (u - 7) : u^{\frac{4}{3}} - 7 3 u

= \frac{1}{4} \cdot \int u^{\frac{4}{3}} - 7 3 u 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}{4} (\int u^{\frac{4}{3}} d u - \int 7 3 u d u)

\int u^{\frac{4}{3}} d u = \frac{3}{7} u^{\frac{7}{3}}

\int 7 3 u d u = \frac{21}{4} u^{\frac{4}{3}}

= \frac{1}{4} (\frac{3}{7} u^{\frac{7}{3}} - \frac{21}{4} u^{\frac{4}{3}})

Setze in

u = 2 t + 7

ein

= \frac{1}{4} (\frac{3}{7} (2 t + 7)^{\frac{7}{3}} - \frac{21}{4} (2 t + 7)^{\frac{4}{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (\frac{3}{7} (2 t + 7)^{\frac{7}{3}} - \frac{21}{4} (2 t + 7)^{\frac{4}{3}}) + C

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