Integralen av $$$- 2 x + 3 \cos{\left(x \right)} + \frac{1}{x}$$$

Bestäm $$$\int \left(- 2 x + 3 \cos{\left(x \right)} + \frac{1}{x}\right)\, dx$$$.

Integrera termvis:

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

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

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

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

$$\ln{\left(\left|{x}\right| \right)} + \int{3 \cos{\left(x \right)} d x} - {\color{red}{\int{2 x d x}}} = \ln{\left(\left|{x}\right| \right)} + \int{3 \cos{\left(x \right)} d x} - {\color{red}{\left(2 \int{x d x}\right)}}$$

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

$$\ln{\left(\left|{x}\right| \right)} + \int{3 \cos{\left(x \right)} d x} - 2 {\color{red}{\int{x d x}}}=\ln{\left(\left|{x}\right| \right)} + \int{3 \cos{\left(x \right)} d x} - 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\ln{\left(\left|{x}\right| \right)} + \int{3 \cos{\left(x \right)} d x} - 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=3$$$ och $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$$- x^{2} + \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{3 \cos{\left(x \right)} d x}}} = - x^{2} + \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(3 \int{\cos{\left(x \right)} d x}\right)}}$$

Integralen av cosinus är $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$- x^{2} + \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\int{\cos{\left(x \right)} d x}}} = - x^{2} + \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\sin{\left(x \right)}}}$$

Alltså,

$$\int{\left(- 2 x + 3 \cos{\left(x \right)} + \frac{1}{x}\right)d x} = - x^{2} + \ln{\left(\left|{x}\right| \right)} + 3 \sin{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(- 2 x + 3 \cos{\left(x \right)} + \frac{1}{x}\right)d x} = - x^{2} + \ln{\left(\left|{x}\right| \right)} + 3 \sin{\left(x \right)}+C$$

$$$\int \left(- 2 x + 3 \cos{\left(x \right)} + \frac{1}{x}\right)\, dx = \left(- x^{2} + \ln\left(\left|{x}\right|\right) + 3 \sin{\left(x \right)}\right) + C$$$A