Funktion $$$- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}$$$ integraali

Määritä $$$\int \left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=-24$$$ ja $$$f{\left(x \right)} = \frac{x}{\left(x - 5\right) \left(x - 3\right)}$$$:

$${\color{red}{\int{\left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)d x}}} = {\color{red}{\left(- 24 \int{\frac{x}{\left(x - 5\right) \left(x - 3\right)} d x}\right)}}$$

Suorita osamurtokehittely (vaiheet voidaan nähdä kohdassa »):

$$- 24 {\color{red}{\int{\frac{x}{\left(x - 5\right) \left(x - 3\right)} d x}}} = - 24 {\color{red}{\int{\left(- \frac{3}{2 \left(x - 3\right)} + \frac{5}{2 \left(x - 5\right)}\right)d x}}}$$

Integroi termi kerrallaan:

$$- 24 {\color{red}{\int{\left(- \frac{3}{2 \left(x - 3\right)} + \frac{5}{2 \left(x - 5\right)}\right)d x}}} = - 24 {\color{red}{\left(\int{\frac{5}{2 \left(x - 5\right)} d x} - \int{\frac{3}{2 \left(x - 3\right)} d x}\right)}}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{3}{2}$$$ ja $$$f{\left(x \right)} = \frac{1}{x - 3}$$$:

$$- 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 24 {\color{red}{\int{\frac{3}{2 \left(x - 3\right)} d x}}} = - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 24 {\color{red}{\left(\frac{3 \int{\frac{1}{x - 3} d x}}{2}\right)}}$$

Olkoon $$$u=x - 3$$$.

Tällöin $$$du=\left(x - 3\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Integraali voidaan kirjoittaa muotoon

$$- 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\int{\frac{1}{x - 3} d x}}} = - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$- 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\int{\frac{1}{u} d u}}} = - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Muista, että $$$u=x - 3$$$:

$$36 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} = 36 \ln{\left(\left|{{\color{red}{\left(x - 3\right)}}}\right| \right)} - 24 \int{\frac{5}{2 \left(x - 5\right)} d x}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{5}{2}$$$ ja $$$f{\left(x \right)} = \frac{1}{x - 5}$$$:

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 24 {\color{red}{\int{\frac{5}{2 \left(x - 5\right)} d x}}} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 24 {\color{red}{\left(\frac{5 \int{\frac{1}{x - 5} d x}}{2}\right)}}$$

Olkoon $$$u=x - 5$$$.

Tällöin $$$du=\left(x - 5\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Siis,

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\int{\frac{1}{x - 5} d x}}} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\int{\frac{1}{u} d u}}} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Muista, että $$$u=x - 5$$$:

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 60 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 60 \ln{\left(\left|{{\color{red}{\left(x - 5\right)}}}\right| \right)}$$

Näin ollen,

$$\int{\left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)d x} = - 60 \ln{\left(\left|{x - 5}\right| \right)} + 36 \ln{\left(\left|{x - 3}\right| \right)}$$

Lisää integrointivakio:

$$\int{\left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)d x} = - 60 \ln{\left(\left|{x - 5}\right| \right)} + 36 \ln{\left(\left|{x - 3}\right| \right)}+C$$

$$$\int \left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)\, dx = \left(- 60 \ln\left(\left|{x - 5}\right|\right) + 36 \ln\left(\left|{x - 3}\right|\right)\right) + C$$$A