Integralen av $$$2 x \ln\left(x^{2}\right)$$$

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

Inmatningen skrivs om: $$$\int{2 x \ln{\left(x^{2} \right)} d x}=\int{4 x \ln{\left(x \right)} d x}$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=4$$$ och $$$f{\left(x \right)} = x \ln{\left(x \right)}$$$:

$${\color{red}{\int{4 x \ln{\left(x \right)} d x}}} = {\color{red}{\left(4 \int{x \ln{\left(x \right)} d x}\right)}}$$

För integralen $$$\int{x \ln{\left(x \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=\ln{\left(x \right)}$$$ och $$$\operatorname{dv}=x dx$$$.

Då gäller $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (stegen kan ses »).

Alltså,

$$4 {\color{red}{\int{x \ln{\left(x \right)} d x}}}=4 {\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{2}}{2}-\int{\frac{x^{2}}{2} \cdot \frac{1}{x} d x}\right)}}=4 {\color{red}{\left(\frac{x^{2} \ln{\left(x \right)}}{2} - \int{\frac{x}{2} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = x$$$:

$$2 x^{2} \ln{\left(x \right)} - 4 {\color{red}{\int{\frac{x}{2} d x}}} = 2 x^{2} \ln{\left(x \right)} - 4 {\color{red}{\left(\frac{\int{x d x}}{2}\right)}}$$

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

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

Alltså,

$$\int{4 x \ln{\left(x \right)} d x} = 2 x^{2} \ln{\left(x \right)} - x^{2}$$

Förenkla:

$$\int{4 x \ln{\left(x \right)} d x} = x^{2} \left(2 \ln{\left(x \right)} - 1\right)$$

Lägg till integrationskonstanten:

$$\int{4 x \ln{\left(x \right)} d x} = x^{2} \left(2 \ln{\left(x \right)} - 1\right)+C$$

$$$\int 2 x \ln\left(x^{2}\right)\, dx = x^{2} \left(2 \ln\left(x\right) - 1\right) + C$$$A