Integralen av $$$\frac{t}{e^{3}}$$$

Bestäm $$$\int \frac{t}{e^{3}}\, dt$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ med $$$c=e^{-3}$$$ och $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{\frac{t}{e^{3}} d t}}} = {\color{red}{\frac{\int{t d t}}{e^{3}}}}$$

Tillämpa potensregeln $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:

$$\frac{{\color{red}{\int{t d t}}}}{e^{3}}=\frac{{\color{red}{\frac{t^{1 + 1}}{1 + 1}}}}{e^{3}}=\frac{{\color{red}{\left(\frac{t^{2}}{2}\right)}}}{e^{3}}$$

Alltså,

$$\int{\frac{t}{e^{3}} d t} = \frac{t^{2}}{2 e^{3}}$$

Lägg till integrationskonstanten:

$$\int{\frac{t}{e^{3}} d t} = \frac{t^{2}}{2 e^{3}}+C$$

$$$\int \frac{t}{e^{3}}\, dt = \frac{t^{2}}{2 e^{3}} + C$$$A