Integraali $$$\frac{i a g h o r^{3} t w \ln^{2}\left(x\right)}{e^{\frac{1}{2}}}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int \frac{i a g h o r^{3} t w \ln^{2}\left(x\right)}{e^{\frac{1}{2}}}\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{i a g h o r^{3} t w}{e^{\frac{1}{2}}}$$$ ja $$$f{\left(x \right)} = \ln{\left(x \right)}^{2}$$$:

$${\color{red}{\int{\frac{i a g h o r^{3} t w \ln{\left(x \right)}^{2}}{e^{\frac{1}{2}}} d x}}} = {\color{red}{\frac{i a g h o r^{3} t w \int{\ln{\left(x \right)}^{2} d x}}{e^{\frac{1}{2}}}}}$$

Integraalin $$$\int{\ln{\left(x \right)}^{2} d x}$$$ kohdalla käytä osittaisintegrointia $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Olkoon $$$\operatorname{u}=\ln{\left(x \right)}^{2}$$$ ja $$$\operatorname{dv}=dx$$$.

Tällöin $$$\operatorname{du}=\left(\ln{\left(x \right)}^{2}\right)^{\prime }dx=\frac{2 \ln{\left(x \right)}}{x} dx$$$ (vaiheet ovat nähtävissä ») ja $$$\operatorname{v}=\int{1 d x}=x$$$ (vaiheet ovat nähtävissä »).

Näin ollen,

$$\frac{i a g h o r^{3} t w {\color{red}{\int{\ln{\left(x \right)}^{2} d x}}}}{e^{\frac{1}{2}}}=\frac{i a g h o r^{3} t w {\color{red}{\left(\ln{\left(x \right)}^{2} \cdot x-\int{x \cdot \frac{2 \ln{\left(x \right)}}{x} d x}\right)}}}{e^{\frac{1}{2}}}=\frac{i a g h o r^{3} t w {\color{red}{\left(x \ln{\left(x \right)}^{2} - \int{2 \ln{\left(x \right)} d x}\right)}}}{e^{\frac{1}{2}}}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=2$$$ ja $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

$$\frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - {\color{red}{\int{2 \ln{\left(x \right)} d x}}}\right)}{e^{\frac{1}{2}}} = \frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - {\color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}}\right)}{e^{\frac{1}{2}}}$$

Integraalin $$$\int{\ln{\left(x \right)} d x}$$$ kohdalla käytä osittaisintegrointia $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

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

Tällöin $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (vaiheet ovat nähtävissä ») ja $$$\operatorname{v}=\int{1 d x}=x$$$ (vaiheet ovat nähtävissä »).

Integraali muuttuu muotoon

$$\frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - 2 {\color{red}{\int{\ln{\left(x \right)} d x}}}\right)}{e^{\frac{1}{2}}}=\frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - 2 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}\right)}{e^{\frac{1}{2}}}=\frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - 2 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}\right)}{e^{\frac{1}{2}}}$$

Sovella vakiosääntöä $$$\int c\, dx = c x$$$ käyttäen $$$c=1$$$:

$$\frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - 2 x \ln{\left(x \right)} + 2 {\color{red}{\int{1 d x}}}\right)}{e^{\frac{1}{2}}} = \frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - 2 x \ln{\left(x \right)} + 2 {\color{red}{x}}\right)}{e^{\frac{1}{2}}}$$

Näin ollen,

$$\int{\frac{i a g h o r^{3} t w \ln{\left(x \right)}^{2}}{e^{\frac{1}{2}}} d x} = \frac{i a g h o r^{3} t w \left(x \ln{\left(x \right)}^{2} - 2 x \ln{\left(x \right)} + 2 x\right)}{e^{\frac{1}{2}}}$$

Sievennä:

$$\int{\frac{i a g h o r^{3} t w \ln{\left(x \right)}^{2}}{e^{\frac{1}{2}}} d x} = \frac{i a g h o r^{3} t w x \left(\ln{\left(x \right)}^{2} - 2 \ln{\left(x \right)} + 2\right)}{e^{\frac{1}{2}}}$$

Lisää integrointivakio:

$$\int{\frac{i a g h o r^{3} t w \ln{\left(x \right)}^{2}}{e^{\frac{1}{2}}} d x} = \frac{i a g h o r^{3} t w x \left(\ln{\left(x \right)}^{2} - 2 \ln{\left(x \right)} + 2\right)}{e^{\frac{1}{2}}}+C$$

$$$\int \frac{i a g h o r^{3} t w \ln^{2}\left(x\right)}{e^{\frac{1}{2}}}\, dx = \frac{i a g h o r^{3} t w x \left(\ln^{2}\left(x\right) - 2 \ln\left(x\right) + 2\right)}{e^{\frac{1}{2}}} + C$$$A