$$$x$$$ değişkenine göre $$$\frac{\sqrt{- a^{2} + x^{2}}}{x}$$$ fonksiyonunun integrali

Bulun: $$$\int \frac{\sqrt{- a^{2} + x^{2}}}{x}\, dx$$$.

$$$x=\cosh{\left(u \right)} \left|{a}\right|$$$ olsun.

O halde $$$dx=\left(\cosh{\left(u \right)} \left|{a}\right|\right)^{\prime }du = \sinh{\left(u \right)} \left|{a}\right| du$$$ (adımlar » görülebilir).

Ayrıca, buradan $$$u=\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}$$$ elde edilir.

İntegrand şu hale gelir

$$$\frac{\sqrt{- a^{2} + x^{2}}}{x} = \frac{\sqrt{a^{2} \cosh^{2}{\left( u \right)} - a^{2}}}{\cosh{\left( u \right)} \left|{a}\right|}$$$

Özdeşliği kullanın: $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$

$$$\frac{\sqrt{a^{2} \cosh^{2}{\left( u \right)} - a^{2}}}{\cosh{\left( u \right)} \left|{a}\right|}=\frac{\sqrt{\cosh^{2}{\left( u \right)} - 1}}{\cosh{\left( u \right)}}=\frac{\sqrt{\sinh^{2}{\left( u \right)}}}{\cosh{\left( u \right)}}$$$

$$$\sinh{\left( u \right)} \ge 0$$$ olduğunu varsayarsak, aşağıdakileri elde ederiz:

$$$\frac{\sqrt{\sinh^{2}{\left( u \right)}}}{\cosh{\left( u \right)}} = \frac{\sinh{\left( u \right)}}{\cosh{\left( u \right)}}$$$

Dolayısıyla,

$${\color{red}{\int{\frac{\sqrt{- a^{2} + x^{2}}}{x} d x}}} = {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \left|{a}\right|}{\cosh{\left(u \right)}} d u}}}$$

Sabit katsayı kuralı $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$'i $$$c=\left|{a}\right|$$$ ve $$$f{\left(u \right)} = \frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}}$$$ ile uygula:

$${\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \left|{a}\right|}{\cosh{\left(u \right)}} d u}}} = {\color{red}{\left|{a}\right| \int{\frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}}}$$

Payı ve paydayı bir hiperbolik kosinüsle çarpın ve geri kalan her şeyi hiperbolik sinüs cinsinden yazın, $$$\cosh^2\left(\alpha \right)=\sinh^2\left(\alpha \right)+1$$$ formülünü $$$\alpha= u $$$ ile kullanarak:

$$\left|{a}\right| {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}}} = \left|{a}\right| {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \cosh{\left(u \right)}}{\sinh^{2}{\left(u \right)} + 1} d u}}}$$

$$$v=\sinh{\left(u \right)}$$$ olsun.

Böylece $$$dv=\left(\sinh{\left(u \right)}\right)^{\prime }du = \cosh{\left(u \right)} du$$$ (adımlar » görülebilir) ve $$$\cosh{\left(u \right)} du = dv$$$ elde ederiz.

İntegral şu hale gelir

$$\left|{a}\right| {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \cosh{\left(u \right)}}{\sinh^{2}{\left(u \right)} + 1} d u}}} = \left|{a}\right| {\color{red}{\int{\frac{v^{2}}{v^{2} + 1} d v}}}$$

Kesri yeniden yazın ve parçalara ayırın:

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

Her terimin integralini alın:

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

$$$c=1$$$ kullanarak $$$\int c\, dv = c v$$$ sabit kuralını uygula:

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

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

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

Hatırlayın ki $$$v=\sinh{\left(u \right)}$$$:

$$\left|{a}\right| \left(- \operatorname{atan}{\left({\color{red}{v}} \right)} + {\color{red}{v}}\right) = \left|{a}\right| \left(- \operatorname{atan}{\left({\color{red}{\sinh{\left(u \right)}}} \right)} + {\color{red}{\sinh{\left(u \right)}}}\right)$$

Hatırlayın ki $$$u=\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}$$$:

$$\left|{a}\right| \left(\sinh{\left({\color{red}{u}} \right)} - \operatorname{atan}{\left(\sinh{\left({\color{red}{u}} \right)} \right)}\right) = \left|{a}\right| \left(\sinh{\left({\color{red}{\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}} \right)} - \operatorname{atan}{\left(\sinh{\left({\color{red}{\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}} \right)} \right)}\right)$$

Dolayısıyla,

$$\int{\frac{\sqrt{- a^{2} + x^{2}}}{x} d x} = \left(\sqrt{\frac{x}{\left|{a}\right|} - 1} \sqrt{\frac{x}{\left|{a}\right|} + 1} - \operatorname{atan}{\left(\sqrt{\frac{x}{\left|{a}\right|} - 1} \sqrt{\frac{x}{\left|{a}\right|} + 1} \right)}\right) \left|{a}\right|$$

İntegrasyon sabitini ekleyin:

$$\int{\frac{\sqrt{- a^{2} + x^{2}}}{x} d x} = \left(\sqrt{\frac{x}{\left|{a}\right|} - 1} \sqrt{\frac{x}{\left|{a}\right|} + 1} - \operatorname{atan}{\left(\sqrt{\frac{x}{\left|{a}\right|} - 1} \sqrt{\frac{x}{\left|{a}\right|} + 1} \right)}\right) \left|{a}\right|+C$$

$$$\int \frac{\sqrt{- a^{2} + x^{2}}}{x}\, dx = \left(\sqrt{\frac{x}{\left|{a}\right|} - 1} \sqrt{\frac{x}{\left|{a}\right|} + 1} - \operatorname{atan}{\left(\sqrt{\frac{x}{\left|{a}\right|} - 1} \sqrt{\frac{x}{\left|{a}\right|} + 1} \right)}\right) \left|{a}\right| + C$$$A