$$$\frac{3 x^{2} - 209 x}{x^{2}}$$$'nin integrali

Bulun: $$$\int \frac{3 x^{2} - 209 x}{x^{2}}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{3 x^{2} - 209 x}{x^{2}} d x}}} = {\color{red}{\int{\left(3 - \frac{209}{x}\right)d x}}}$$

Her terimin integralini alın:

$${\color{red}{\int{\left(3 - \frac{209}{x}\right)d x}}} = {\color{red}{\left(\int{3 d x} - \int{\frac{209}{x} d x}\right)}}$$

$$$c=3$$$ kullanarak $$$\int c\, dx = c x$$$ sabit kuralını uygula:

$$- \int{\frac{209}{x} d x} + {\color{red}{\int{3 d x}}} = - \int{\frac{209}{x} d x} + {\color{red}{\left(3 x\right)}}$$

Sabit katsayı kuralı $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$'i $$$c=209$$$ ve $$$f{\left(x \right)} = \frac{1}{x}$$$ ile uygula:

$$3 x - {\color{red}{\int{\frac{209}{x} d x}}} = 3 x - {\color{red}{\left(209 \int{\frac{1}{x} d x}\right)}}$$

$$$\frac{1}{x}$$$'nin integrali $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$3 x - 209 {\color{red}{\int{\frac{1}{x} d x}}} = 3 x - 209 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Dolayısıyla,

$$\int{\frac{3 x^{2} - 209 x}{x^{2}} d x} = 3 x - 209 \ln{\left(\left|{x}\right| \right)}$$

İntegrasyon sabitini ekleyin:

$$\int{\frac{3 x^{2} - 209 x}{x^{2}} d x} = 3 x - 209 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{3 x^{2} - 209 x}{x^{2}}\, dx = \left(3 x - 209 \ln\left(\left|{x}\right|\right)\right) + C$$$A