Integralen av $$$\frac{5 x}{e^{\frac{1}{10}}}$$$

Bestäm $$$\int \frac{5 x}{e^{\frac{1}{10}}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{5}{e^{\frac{1}{10}}}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{5 x}{e^{\frac{1}{10}}} d x}}} = {\color{red}{\left(\frac{5 \int{x d x}}{e^{\frac{1}{10}}}\right)}}$$

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

$$\frac{5 {\color{red}{\int{x d x}}}}{e^{\frac{1}{10}}}=\frac{5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{e^{\frac{1}{10}}}=\frac{5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{e^{\frac{1}{10}}}$$

Alltså,

$$\int{\frac{5 x}{e^{\frac{1}{10}}} d x} = \frac{5 x^{2}}{2 e^{\frac{1}{10}}}$$

Lägg till integrationskonstanten:

$$\int{\frac{5 x}{e^{\frac{1}{10}}} d x} = \frac{5 x^{2}}{2 e^{\frac{1}{10}}}+C$$

$$$\int \frac{5 x}{e^{\frac{1}{10}}}\, dx = \frac{5 x^{2}}{2 e^{\frac{1}{10}}} + C$$$A