$$$\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$$'nin integrali

Bulun: $$$\int \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx$$$.

Kotanjant cinsinden yeniden yazın:

$${\color{red}{\int{\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\cot^{2}{\left(x \right)} d x}}}$$

$$$u=\cot{\left(x \right)}$$$ olsun.

Böylece $$$du=\left(\cot{\left(x \right)}\right)^{\prime }dx = - \csc^{2}{\left(x \right)} dx$$$ (adımlar » görülebilir) ve $$$\csc^{2}{\left(x \right)} dx = - du$$$ elde ederiz.

O halde,

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

Sabit katsayı kuralı $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$'i $$$c=-1$$$ ve $$$f{\left(u \right)} = \frac{u^{2}}{u^{2} + 1}$$$ ile uygula:

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

Kesri yeniden yazın ve parçalara ayırın:

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

Her terimin integralini alın:

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

$$$c=1$$$ kullanarak $$$\int c\, du = c u$$$ sabit kuralını uygula:

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

$$$\frac{1}{u^{2} + 1}$$$'nin integrali $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

$$- u + {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}} = - u + {\color{red}{\operatorname{atan}{\left(u \right)}}}$$

Hatırlayın ki $$$u=\cot{\left(x \right)}$$$:

$$\operatorname{atan}{\left({\color{red}{u}} \right)} - {\color{red}{u}} = \operatorname{atan}{\left({\color{red}{\cot{\left(x \right)}}} \right)} - {\color{red}{\cot{\left(x \right)}}}$$

Dolayısıyla,

$$\int{\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} d x} = - \cot{\left(x \right)} + \operatorname{atan}{\left(\cot{\left(x \right)} \right)}$$

İntegrasyon sabitini ekleyin:

$$\int{\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} d x} = - \cot{\left(x \right)} + \operatorname{atan}{\left(\cot{\left(x \right)} \right)}+C$$

$$$\int \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx = \left(- \cot{\left(x \right)} + \operatorname{atan}{\left(\cot{\left(x \right)} \right)}\right) + C$$$A