integral from 0 to 1 of 4/3 t^{3/2}(t+2)

Lösung

\int_{0}^{1} \frac{4}{3} t^{\frac{3}{2}} (t + 2) d t

\frac{152}{105}

Dezimale

1.44761 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{4}{3} t^{\frac{3}{2}} (t + 2) d t

Entferne die Konstante:

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

= \frac{4}{3} \cdot \int_{0}^{1} t^{\frac{3}{2}} (t + 2) d t

Wende die partielle Integration an

= \frac{4}{3} [\frac{2}{5} t^{\frac{5}{2}} (t + 2) - \int \frac{2}{5} t^{\frac{5}{2}} d t]_{0}^{1}

\int \frac{2}{5} t^{\frac{5}{2}} d t = \frac{4}{35} t^{\frac{7}{2}}

= \frac{4}{3} [\frac{2}{5} t^{\frac{5}{2}} (t + 2) - \frac{4}{35} t^{\frac{7}{2}}]_{0}^{1}

Vereinfache

\frac{4}{3} [\frac{2}{5} t^{\frac{5}{2}} (t + 2) - \frac{4}{35} t^{\frac{7}{2}}]_{0}^{1} : \frac{4}{3} [\frac{10 t ^{\frac{7}{2}} + 28 t ^{\frac{5}{2}}}{35}]_{0}^{1}

= \frac{4}{3} [\frac{10 t ^{\frac{7}{2}} + 28 t ^{\frac{5}{2}}}{35}]_{0}^{1}

Berechne die Grenzen:

\frac{38}{35}

= \frac{4}{3} \cdot \frac{38}{35}

Vereinfache

= \frac{152}{105}

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