Funktion $$$\frac{1}{x^{2} - 16}$$$ integraali

Määritä $$$\int \frac{1}{x^{2} - 16}\, dx$$$.

Suorita osamurtokehittely (vaiheet voidaan nähdä kohdassa »):

$${\color{red}{\int{\frac{1}{x^{2} - 16} d x}}} = {\color{red}{\int{\left(- \frac{1}{8 \left(x + 4\right)} + \frac{1}{8 \left(x - 4\right)}\right)d x}}}$$

Integroi termi kerrallaan:

$${\color{red}{\int{\left(- \frac{1}{8 \left(x + 4\right)} + \frac{1}{8 \left(x - 4\right)}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{8 \left(x - 4\right)} d x} - \int{\frac{1}{8 \left(x + 4\right)} d x}\right)}}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{1}{8}$$$ ja $$$f{\left(x \right)} = \frac{1}{x + 4}$$$:

$$\int{\frac{1}{8 \left(x - 4\right)} d x} - {\color{red}{\int{\frac{1}{8 \left(x + 4\right)} d x}}} = \int{\frac{1}{8 \left(x - 4\right)} d x} - {\color{red}{\left(\frac{\int{\frac{1}{x + 4} d x}}{8}\right)}}$$

Olkoon $$$u=x + 4$$$.

Tällöin $$$du=\left(x + 4\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Integraali muuttuu muotoon

$$\int{\frac{1}{8 \left(x - 4\right)} d x} - \frac{{\color{red}{\int{\frac{1}{x + 4} d x}}}}{8} = \int{\frac{1}{8 \left(x - 4\right)} d x} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{8}$$

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

$$\int{\frac{1}{8 \left(x - 4\right)} d x} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{8} = \int{\frac{1}{8 \left(x - 4\right)} d x} - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{8}$$

Muista, että $$$u=x + 4$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{8} + \int{\frac{1}{8 \left(x - 4\right)} d x} = - \frac{\ln{\left(\left|{{\color{red}{\left(x + 4\right)}}}\right| \right)}}{8} + \int{\frac{1}{8 \left(x - 4\right)} d x}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{1}{8}$$$ ja $$$f{\left(x \right)} = \frac{1}{x - 4}$$$:

$$- \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + {\color{red}{\int{\frac{1}{8 \left(x - 4\right)} d x}}} = - \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + {\color{red}{\left(\frac{\int{\frac{1}{x - 4} d x}}{8}\right)}}$$

Olkoon $$$u=x - 4$$$.

Tällöin $$$du=\left(x - 4\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Siis,

$$- \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + \frac{{\color{red}{\int{\frac{1}{x - 4} d x}}}}{8} = - \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{8}$$

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

$$- \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{8} = - \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{8}$$

Muista, että $$$u=x - 4$$$:

$$- \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{8} = - \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8} + \frac{\ln{\left(\left|{{\color{red}{\left(x - 4\right)}}}\right| \right)}}{8}$$

Näin ollen,

$$\int{\frac{1}{x^{2} - 16} d x} = \frac{\ln{\left(\left|{x - 4}\right| \right)}}{8} - \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8}$$

Lisää integrointivakio:

$$\int{\frac{1}{x^{2} - 16} d x} = \frac{\ln{\left(\left|{x - 4}\right| \right)}}{8} - \frac{\ln{\left(\left|{x + 4}\right| \right)}}{8}+C$$

$$$\int \frac{1}{x^{2} - 16}\, dx = \left(\frac{\ln\left(\left|{x - 4}\right|\right)}{8} - \frac{\ln\left(\left|{x + 4}\right|\right)}{8}\right) + C$$$A