Integralen av $$$\ln\left(x \sqrt{x^{21}}\right)$$$

Bestäm $$$\int \ln\left(x \sqrt{x^{21}}\right)\, dx$$$.

Inmatningen skrivs om: $$$\int{\ln{\left(x \sqrt{x^{21}} \right)} d x}=\int{\frac{23 \ln{\left(x \right)}}{2} d x}$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{23}{2}$$$ och $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

$${\color{red}{\int{\frac{23 \ln{\left(x \right)}}{2} d x}}} = {\color{red}{\left(\frac{23 \int{\ln{\left(x \right)} d x}}{2}\right)}}$$

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

Då gäller $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{1 d x}=x$$$ (stegen kan ses »).

Integralen blir

$$\frac{23 {\color{red}{\int{\ln{\left(x \right)} d x}}}}{2}=\frac{23 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}}{2}=\frac{23 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}}{2}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\frac{23 x \ln{\left(x \right)}}{2} - \frac{23 {\color{red}{\int{1 d x}}}}{2} = \frac{23 x \ln{\left(x \right)}}{2} - \frac{23 {\color{red}{x}}}{2}$$

Alltså,

$$\int{\frac{23 \ln{\left(x \right)}}{2} d x} = \frac{23 x \ln{\left(x \right)}}{2} - \frac{23 x}{2}$$

Förenkla:

$$\int{\frac{23 \ln{\left(x \right)}}{2} d x} = \frac{23 x \left(\ln{\left(x \right)} - 1\right)}{2}$$

Lägg till integrationskonstanten:

$$\int{\frac{23 \ln{\left(x \right)}}{2} d x} = \frac{23 x \left(\ln{\left(x \right)} - 1\right)}{2}+C$$

$$$\int \ln\left(x \sqrt{x^{21}}\right)\, dx = \frac{23 x \left(\ln\left(x\right) - 1\right)}{2} + C$$$A