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

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

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

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

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

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 »).

Integralen blir

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

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

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

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

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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