Integraal van $$$\frac{25 x}{e^{\frac{1}{20}}}$$$

Bepaal $$$\int \frac{25 x}{e^{\frac{1}{20}}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{25}{e^{\frac{1}{20}}}$$$ en $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{25 x}{e^{\frac{1}{20}}} d x}}} = {\color{red}{\left(\frac{25 \int{x d x}}{e^{\frac{1}{20}}}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

$$\frac{25 {\color{red}{\int{x d x}}}}{e^{\frac{1}{20}}}=\frac{25 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{e^{\frac{1}{20}}}=\frac{25 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{e^{\frac{1}{20}}}$$

Dus,

$$\int{\frac{25 x}{e^{\frac{1}{20}}} d x} = \frac{25 x^{2}}{2 e^{\frac{1}{20}}}$$

Voeg de integratieconstante toe:

$$\int{\frac{25 x}{e^{\frac{1}{20}}} d x} = \frac{25 x^{2}}{2 e^{\frac{1}{20}}}+C$$

$$$\int \frac{25 x}{e^{\frac{1}{20}}}\, dx = \frac{25 x^{2}}{2 e^{\frac{1}{20}}} + C$$$A