Integraali $$$\frac{x^{3}}{m^{2} + 4 x^{2}}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int \frac{x^{3}}{m^{2} + 4 x^{2}}\, dx$$$.

Koska osoittajan aste ei ole pienempi kuin nimittäjän aste, suorita polynomien jakokulma.:

$${\color{red}{\int{\frac{x^{3}}{m^{2} + 4 x^{2}} d x}}} = {\color{red}{\int{\left(- \frac{m^{2} x}{4 \left(m^{2} + 4 x^{2}\right)} + \frac{x}{4}\right)d x}}}$$

Integroi termi kerrallaan:

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

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

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

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

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

Olkoon $$$u=4 m^{2} + 16 x^{2}$$$.

Tällöin $$$du=\left(4 m^{2} + 16 x^{2}\right)^{\prime }dx = 32 x dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$x dx = \frac{du}{32}$$$.

Integraali muuttuu muotoon

$$\frac{x^{2}}{8} - {\color{red}{\int{\frac{m^{2} x}{4 \left(m^{2} + 4 x^{2}\right)} d x}}} = \frac{x^{2}}{8} - {\color{red}{\int{\frac{m^{2}}{32 u} d u}}}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ käyttäen $$$c=\frac{m^{2}}{32}$$$ ja $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$\frac{x^{2}}{8} - {\color{red}{\int{\frac{m^{2}}{32 u} d u}}} = \frac{x^{2}}{8} - {\color{red}{\left(\frac{m^{2} \int{\frac{1}{u} d u}}{32}\right)}}$$

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

$$- \frac{m^{2} {\color{red}{\int{\frac{1}{u} d u}}}}{32} + \frac{x^{2}}{8} = - \frac{m^{2} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{32} + \frac{x^{2}}{8}$$

Muista, että $$$u=4 m^{2} + 16 x^{2}$$$:

$$- \frac{m^{2} \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{32} + \frac{x^{2}}{8} = - \frac{m^{2} \ln{\left(\left|{{\color{red}{\left(4 m^{2} + 16 x^{2}\right)}}}\right| \right)}}{32} + \frac{x^{2}}{8}$$

Näin ollen,

$$\int{\frac{x^{3}}{m^{2} + 4 x^{2}} d x} = - \frac{m^{2} \ln{\left(4 m^{2} + 16 x^{2} \right)}}{32} + \frac{x^{2}}{8}$$

Sievennä:

$$\int{\frac{x^{3}}{m^{2} + 4 x^{2}} d x} = - \frac{m^{2} \left(\ln{\left(m^{2} + 4 x^{2} \right)} + 2 \ln{\left(2 \right)}\right)}{32} + \frac{x^{2}}{8}$$

Lisää integrointivakio:

$$\int{\frac{x^{3}}{m^{2} + 4 x^{2}} d x} = - \frac{m^{2} \left(\ln{\left(m^{2} + 4 x^{2} \right)} + 2 \ln{\left(2 \right)}\right)}{32} + \frac{x^{2}}{8}+C$$

$$$\int \frac{x^{3}}{m^{2} + 4 x^{2}}\, dx = \left(- \frac{m^{2} \left(\ln\left(m^{2} + 4 x^{2}\right) + 2 \ln\left(2\right)\right)}{32} + \frac{x^{2}}{8}\right) + C$$$A