Integraali $$$\frac{1}{a^{2} - x^{2}}$$$:stä muuttujan $$$x$$$ suhteen

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

Hajota osamurtoihin:

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

Integroi termi kerrallaan:

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

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

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

Olkoon $$$u=a + x$$$.

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

Näin ollen,

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

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

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

Muista, että $$$u=a + x$$$:

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

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

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

Olkoon $$$u=- a + x$$$.

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

Näin ollen,

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

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

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

Muista, että $$$u=- a + x$$$:

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

Näin ollen,

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

Sievennä:

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

Lisää integrointivakio:

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

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