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Beliebt Rechnen >

integral of (t+5)^2sqrt(3-t)

Lösung

\int (t + 5)^{2} 3 - t d t

Lösung

- \frac{2}{7} (- t + 3)^{\frac{7}{2}} + \frac{32}{5} (- t + 3)^{\frac{5}{2}} - \frac{128}{3} (- t + 3)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (t + 5)^{2} 3 - t d t

Multipliziere aus

(t + 5)^{2} 3 - t : t^{2} 3 - t + 10 t 3 - t + 25 3 - t

= \int t^{2} 3 - t + 10 t 3 - t + 25 3 - t d t

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int t^{2} 3 - t d t + \int 10 t 3 - t d t + \int 25 3 - t d t

\int t^{2} 3 - t d t = - \frac{2}{7} (3 - t)^{\frac{7}{2}} + \frac{12}{5} (3 - t)^{\frac{5}{2}} - 6 (3 - t)^{\frac{3}{2}}

\int 10 t 3 - t d t = - 10 (- \frac{2}{5} (3 - t)^{\frac{5}{2}} + 2 (3 - t)^{\frac{3}{2}})

\int 25 3 - t d t = - \frac{50}{3} (3 - t)^{\frac{3}{2}}

= - \frac{2}{7} (3 - t)^{\frac{7}{2}} + \frac{12}{5} (3 - t)^{\frac{5}{2}} - 6 (3 - t)^{\frac{3}{2}} - 10 (- \frac{2}{5} (3 - t)^{\frac{5}{2}} + 2 (3 - t)^{\frac{3}{2}}) - \frac{50}{3} (3 - t)^{\frac{3}{2}}

Vereinfache

- \frac{2}{7} (3 - t)^{\frac{7}{2}} + \frac{12}{5} (3 - t)^{\frac{5}{2}} - 6 (3 - t)^{\frac{3}{2}} - 10 (- \frac{2}{5} (3 - t)^{\frac{5}{2}} + 2 (3 - t)^{\frac{3}{2}}) - \frac{50}{3} (3 - t)^{\frac{3}{2}} : - \frac{2}{7} (- t + 3)^{\frac{7}{2}} + \frac{32}{5} (- t + 3)^{\frac{5}{2}} - \frac{128}{3} (- t + 3)^{\frac{3}{2}}

= - \frac{2}{7} (- t + 3)^{\frac{7}{2}} + \frac{32}{5} (- t + 3)^{\frac{5}{2}} - \frac{128}{3} (- t + 3)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{7} (- t + 3)^{\frac{7}{2}} + \frac{32}{5} (- t + 3)^{\frac{5}{2}} - \frac{128}{3} (- t + 3)^{\frac{3}{2}} + C

Graph

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