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Beliebt Rechnen >

integral of t(3t+2)^{3/2}

Lösung

\int t (3 t + 2)^{\frac{3}{2}} d t

Lösung

\frac{1}{9} (\frac{2}{7} (3 t + 2)^{\frac{7}{2}} - \frac{4}{5} (3 t + 2)^{\frac{5}{2}}) + C

Schritte zur Lösung

\int t (3 t + 2)^{\frac{3}{2}} d t

Wende U-Substitution an

= \int \frac{u ^{\frac{3}{2}} ( u - 2 )}{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^{\frac{3}{2}} (u - 2) d u

Multipliziere aus

u^{\frac{3}{2}} (u - 2) : u^{\frac{5}{2}} - 2 u^{\frac{3}{2}}

= \frac{1}{9} \cdot \int u^{\frac{5}{2}} - 2 u^{\frac{3}{2}} 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^{\frac{5}{2}} d u - \int 2 u^{\frac{3}{2}} d u)

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

\int 2 u^{\frac{3}{2}} d u = \frac{4}{5} u^{\frac{5}{2}}

= \frac{1}{9} (\frac{2}{7} u^{\frac{7}{2}} - \frac{4}{5} u^{\frac{5}{2}})

Setze in

u = 3 t + 2

ein

= \frac{1}{9} (\frac{2}{7} (3 t + 2)^{\frac{7}{2}} - \frac{4}{5} (3 t + 2)^{\frac{5}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} (\frac{2}{7} (3 t + 2)^{\frac{7}{2}} - \frac{4}{5} (3 t + 2)^{\frac{5}{2}}) + C

Graph

Beliebte Beispiele

integral of-2sin(2t+7)

\int - 2 sin (2 t + 7) d t

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\int_{0}^{3} x^{4} d x

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limit as x approaches 4 of x^2+5x

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