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

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

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

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

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

För integralen $$$\int{x^{2} \ln{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=x^{2} dx$$$.

Då gäller $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{x^{2} d x}=\frac{x^{3}}{3}$$$ (stegen kan ses »).

Integralen kan omskrivas som

$$2 {\color{red}{\int{x^{2} \ln{\left(x \right)} d x}}}=2 {\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{3}}{3}-\int{\frac{x^{3}}{3} \cdot \frac{1}{x} d x}\right)}}=2 {\color{red}{\left(\frac{x^{3} \ln{\left(x \right)}}{3} - \int{\frac{x^{2}}{3} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$$\frac{2 x^{3} \ln{\left(x \right)}}{3} - 2 {\color{red}{\int{\frac{x^{2}}{3} d x}}} = \frac{2 x^{3} \ln{\left(x \right)}}{3} - 2 {\color{red}{\left(\frac{\int{x^{2} d x}}{3}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

$$\frac{2 x^{3} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\int{x^{2} d x}}}}{3}=\frac{2 x^{3} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{3}=\frac{2 x^{3} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{3}$$

Alltså,

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

Förenkla:

$$\int{2 x^{2} \ln{\left(x \right)} d x} = \frac{2 x^{3} \left(3 \ln{\left(x \right)} - 1\right)}{9}$$

Lägg till integrationskonstanten:

$$\int{2 x^{2} \ln{\left(x \right)} d x} = \frac{2 x^{3} \left(3 \ln{\left(x \right)} - 1\right)}{9}+C$$

$$$\int x^{2} \ln\left(x^{2}\right)\, dx = \frac{2 x^{3} \left(3 \ln\left(x\right) - 1\right)}{9} + C$$$A