Funktion $$$\frac{1}{\left(x - 2\right) \left(x - 1\right)^{2}}$$$ integraali

Määritä $$$\int \frac{1}{\left(x - 2\right) \left(x - 1\right)^{2}}\, dx$$$.

Suorita osamurtokehittely (vaiheet voidaan nähdä kohdassa »):

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

Integroi termi kerrallaan:

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

Olkoon $$$u=x - 2$$$.

Tällöin $$$du=\left(x - 2\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Näin ollen,

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

Funktion $$$\frac{1}{u}$$$ integraali on $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Muista, että $$$u=x - 2$$$:

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

Olkoon $$$u=x - 1$$$.

Tällöin $$$du=\left(x - 1\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Integraali voidaan kirjoittaa muotoon

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

Funktion $$$\frac{1}{u}$$$ integraali on $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Muista, että $$$u=x - 1$$$:

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

Olkoon $$$u=x - 1$$$.

Tällöin $$$du=\left(x - 1\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Näin ollen,

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

Sovella potenssisääntöä $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=-2$$$:

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

Muista, että $$$u=x - 1$$$:

$$\ln{\left(\left|{x - 2}\right| \right)} - \ln{\left(\left|{x - 1}\right| \right)} + {\color{red}{u}}^{-1} = \ln{\left(\left|{x - 2}\right| \right)} - \ln{\left(\left|{x - 1}\right| \right)} + {\color{red}{\left(x - 1\right)}}^{-1}$$

Näin ollen,

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

Sievennä:

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

Lisää integrointivakio:

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

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