Integralen av $$$\frac{x + 4}{x^{2} + 4 x + 4}$$$

Bestäm $$$\int \frac{x + 4}{x^{2} + 4 x + 4}\, dx$$$.

Skriv om den linjära termen som $$$x + 4=x\color{red}{+2-2}+4=x+2+2$$$ och dela upp uttrycket:

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

Integrera termvis:

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

Låt $$$u=x^{2} + 4 x + 4$$$ vara.

Då $$$du=\left(x^{2} + 4 x + 4\right)^{\prime }dx = \left(2 x + 4\right) dx$$$ (stegen kan ses »), och vi har att $$$\left(2 x + 4\right) dx = du$$$.

Integralen kan omskrivas som

$$\int{\frac{2}{x^{2} + 4 x + 4} d x} + {\color{red}{\int{\frac{x + 2}{x^{2} + 4 x + 4} d x}}} = \int{\frac{2}{x^{2} + 4 x + 4} d x} + {\color{red}{\int{\frac{1}{2 u} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \frac{1}{u}$$$:

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

Integralen av $$$\frac{1}{u}$$$ är $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Kom ihåg att $$$u=x^{2} + 4 x + 4$$$:

$$\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} + \int{\frac{2}{x^{2} + 4 x + 4} d x} = \frac{\ln{\left(\left|{{\color{red}{\left(x^{2} + 4 x + 4\right)}}}\right| \right)}}{2} + \int{\frac{2}{x^{2} + 4 x + 4} d x}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \frac{1}{x^{2} + 4 x + 4}$$$:

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

Kvadratkomplettera (stegen kan ses »): $$$x^{2} + 4 x + 4 = \left(x + 2\right)^{2}$$$:

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

Låt $$$u=x + 2$$$ vara.

Då $$$du=\left(x + 2\right)^{\prime }dx = 1 dx$$$ (stegen kan ses »), och vi har att $$$dx = du$$$.

Alltså,

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

Tillämpa potensregeln $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-2$$$:

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

Kom ihåg att $$$u=x + 2$$$:

$$\frac{\ln{\left(\left|{x^{2} + 4 x + 4}\right| \right)}}{2} - 2 {\color{red}{u}}^{-1} = \frac{\ln{\left(\left|{x^{2} + 4 x + 4}\right| \right)}}{2} - 2 {\color{red}{\left(x + 2\right)}}^{-1}$$

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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