Integralen av $$$x \ln\left(2\right)$$$

Bestäm $$$\int x \ln\left(2\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\ln{\left(2 \right)}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{x \ln{\left(2 \right)} d x}}} = {\color{red}{\ln{\left(2 \right)} \int{x d x}}}$$

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

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

Alltså,

$$\int{x \ln{\left(2 \right)} d x} = \frac{x^{2} \ln{\left(2 \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{x \ln{\left(2 \right)} d x} = \frac{x^{2} \ln{\left(2 \right)}}{2}+C$$

$$$\int x \ln\left(2\right)\, dx = \frac{x^{2} \ln\left(2\right)}{2} + C$$$A