Integralen av $$$2 x \ln\left(9\right)$$$

Bestäm $$$\int 2 x \ln\left(9\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2 \ln{\left(9 \right)}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{2 x \ln{\left(9 \right)} d x}}} = {\color{red}{\left(2 \ln{\left(9 \right)} \int{x d x}\right)}}$$

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

$$2 \ln{\left(9 \right)} {\color{red}{\int{x d x}}}=2 \ln{\left(9 \right)} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=2 \ln{\left(9 \right)} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{2 x \ln{\left(9 \right)} d x} = x^{2} \ln{\left(9 \right)}$$

Förenkla:

$$\int{2 x \ln{\left(9 \right)} d x} = 2 x^{2} \ln{\left(3 \right)}$$

Lägg till integrationskonstanten:

$$\int{2 x \ln{\left(9 \right)} d x} = 2 x^{2} \ln{\left(3 \right)}+C$$

$$$\int 2 x \ln\left(9\right)\, dx = 2 x^{2} \ln\left(3\right) + C$$$A