Integralen av $$$\frac{x^{15} \ln\left(x^{16}\right)}{16}$$$

Bestäm $$$\int \frac{x^{15} \ln\left(x^{16}\right)}{16}\, dx$$$.

Inmatningen skrivs om: $$$\int{\frac{x^{15} \ln{\left(x^{16} \right)}}{16} d x}=\int{x^{15} \ln{\left(x \right)} d x}$$$.

För integralen $$$\int{x^{15} \ln{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=x^{15} dx$$$.

Då gäller $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{x^{15} d x}=\frac{x^{16}}{16}$$$ (stegen kan ses »).

Alltså,

$${\color{red}{\int{x^{15} \ln{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{16}}{16}-\int{\frac{x^{16}}{16} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\frac{x^{16} \ln{\left(x \right)}}{16} - \int{\frac{x^{15}}{16} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{16}$$$ och $$$f{\left(x \right)} = x^{15}$$$:

$$\frac{x^{16} \ln{\left(x \right)}}{16} - {\color{red}{\int{\frac{x^{15}}{16} d x}}} = \frac{x^{16} \ln{\left(x \right)}}{16} - {\color{red}{\left(\frac{\int{x^{15} d x}}{16}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=15$$$:

$$\frac{x^{16} \ln{\left(x \right)}}{16} - \frac{{\color{red}{\int{x^{15} d x}}}}{16}=\frac{x^{16} \ln{\left(x \right)}}{16} - \frac{{\color{red}{\frac{x^{1 + 15}}{1 + 15}}}}{16}=\frac{x^{16} \ln{\left(x \right)}}{16} - \frac{{\color{red}{\left(\frac{x^{16}}{16}\right)}}}{16}$$

Alltså,

$$\int{x^{15} \ln{\left(x \right)} d x} = \frac{x^{16} \ln{\left(x \right)}}{16} - \frac{x^{16}}{256}$$

Förenkla:

$$\int{x^{15} \ln{\left(x \right)} d x} = \frac{x^{16} \left(16 \ln{\left(x \right)} - 1\right)}{256}$$

Lägg till integrationskonstanten:

$$\int{x^{15} \ln{\left(x \right)} d x} = \frac{x^{16} \left(16 \ln{\left(x \right)} - 1\right)}{256}+C$$

$$$\int \frac{x^{15} \ln\left(x^{16}\right)}{16}\, dx = \frac{x^{16} \left(16 \ln\left(x\right) - 1\right)}{256} + C$$$A