integral from 0 to t of t^2e^{-t/3}

Lösung

\int_{0}^{t} t^{2} e^{- \frac{t}{3}} d t

- 3 e^{- \frac{t}{3}} t^{2} + 6 (- 3 e^{- \frac{t}{3}} t - 9 e^{- \frac{t}{3}}) + 54

Schritte zur Lösung

\int_{0}^{t} t^{2} e^{- \frac{t}{3}} d t

Wende die partielle Integration an

= [- 3 e^{- \frac{t}{3}} t^{2} - \int - 6 e^{- \frac{t}{3}} t d t]_{0}^{t}

\int - 6 e^{- \frac{t}{3}} t d t = - 6 (- 3 e^{- \frac{t}{3}} t - 9 e^{- \frac{t}{3}})

= [- 3 e^{- \frac{t}{3}} t^{2} - (- 6 (- 3 e^{- \frac{t}{3}} t - 9 e^{- \frac{t}{3}}))]_{0}^{t}

Vereinfache

= [- 3 e^{- \frac{t}{3}} t^{2} + 6 (- 3 e^{- \frac{t}{3}} t - 9 e^{- \frac{t}{3}})]_{0}^{t}

Berechne die Grenzen:

- 3 e^{- \frac{t}{3}} t^{2} + 6 (- 3 e^{- \frac{t}{3}} t - 9 e^{- \frac{t}{3}}) + 54

= - 3 e^{- \frac{t}{3}} t^{2} + 6 (- 3 e^{- \frac{t}{3}} t - 9 e^{- \frac{t}{3}}) + 54

t an g e n t y = \frac{5}{x}, (5, 1)

\int_{1}^{8} (\frac{2}{x}) d x

t an g e n t f (x) = x^{3}, at x = 3

\int - 2 x e^{- \frac{1}{2} x} d x

\frac{\partial}{\partial x} (cos (x^{2}))