aufgabe 1a

$$\int t^2{e}^{t}dt=\dots$$

5.10 Partielle Integration

$$\int f(x)\cdot g^{\prime }(x)dx=f(x)\cdot g(x)-\int f^{\prime }(x)\cdot g(x)dx$$

Nebenrechnung

$$\begin{array}{cc}f(t)=t^2& f^{\prime }(t)=2t\\ g(t)=e^t& g^{\prime }(t)=e^t\end{array}$$

$$\dots =t^2\cdot e^t-\int 2t\cdot e^t$$

Nebenrechnung

$$\begin{array}{cc}f(t)=2t& f^{\prime }(t)=2\\ g(t)=e^t& g^{\prime }(t)=e^t\end{array}$$

$$\dots =t^2\cdot e^t-\left(2t\cdot e^t-\int 2\cdot e^t\right)$$

Nebenrechnung

$$\begin{array}{cc}f(t)=2& f^{\prime }(t)=0\\ g(t)=e^t& g^{\prime }(t)=e^t\end{array}$$

$$\begin{array}{rl}\dots & =t^2\cdot e^t-\left(2t\cdot e^t-\left(2\cdot e^t-\int 0\cdot e^t\right)\right)\\ & =t^2\cdot e^t-2t\cdot e^t+2\cdot e^t-0+c\\ & =\underline{e^t\cdot (t^2-2t+2)+c}\end{array}$$

+c nicht vergessen