integral of t^3(25+81t^2)^{5/2}

Lösung

\int t^{3} (25 + 81 t^{2})^{\frac{5}{2}} d t

\frac{1}{413343} (567 t^{2} - 50) (81 t^{2} + 25)^{\frac{7}{2}} + C

Schritte zur Lösung

\int t^{3} (25 + 81 t^{2})^{\frac{5}{2}} d t

Wende U-Substitution an

= \int \frac{u ^{\frac{5}{2}} ( u - 25 )}{13122} d u

Entferne die Konstante:

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

= \frac{1}{13122} \cdot \int u^{\frac{5}{2}} (u - 25) d u

Multipliziere aus

u^{\frac{5}{2}} (u - 25) : u^{\frac{7}{2}} - 25 u^{\frac{5}{2}}

= \frac{1}{13122} \cdot \int u^{\frac{7}{2}} - 25 u^{\frac{5}{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}{13122} (\int u^{\frac{7}{2}} d u - \int 25 u^{\frac{5}{2}} d u)

\int u^{\frac{7}{2}} d u = \frac{2}{9} u^{\frac{9}{2}}

\int 25 u^{\frac{5}{2}} d u = \frac{50}{7} u^{\frac{7}{2}}

= \frac{1}{13122} (\frac{2}{9} u^{\frac{9}{2}} - \frac{50}{7} u^{\frac{7}{2}})

Setze in

u = 25 + 81 t^{2}

ein

= \frac{1}{13122} (\frac{2}{9} (25 + 81 t^{2})^{\frac{9}{2}} - \frac{50}{7} (25 + 81 t^{2})^{\frac{7}{2}})

Vereinfache

\frac{1}{13122} (\frac{2}{9} (25 + 81 t^{2})^{\frac{9}{2}} - \frac{50}{7} (25 + 81 t^{2})^{\frac{7}{2}}) : \frac{1}{413343} (567 t^{2} - 50) (81 t^{2} + 25)^{\frac{7}{2}}

= \frac{1}{413343} (567 t^{2} - 50) (81 t^{2} + 25)^{\frac{7}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{413343} (567 t^{2} - 50) (81 t^{2} + 25)^{\frac{7}{2}} + C

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