Integral dari $$$\frac{1}{x \sqrt{x^{2} - 1}}$$$

Temukan $$$\int \frac{1}{x \sqrt{x^{2} - 1}}\, dx$$$.

Misalkan $$$u=\frac{1}{x}$$$.

Kemudian $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\frac{dx}{x^{2}} = - du$$$.

Dengan demikian,

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=-1$$$ dan $$$f{\left(u \right)} = \frac{1}{\sqrt{1 - u^{2}}}$$$:

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

Misalkan $$$u=\sin{\left(v \right)}$$$.

Maka $$$du=\left(\sin{\left(v \right)}\right)^{\prime }dv = \cos{\left(v \right)} dv$$$ (langkah-langkah dapat dilihat »).

Selain itu, berlaku $$$v=\operatorname{asin}{\left(u \right)}$$$.

Jadi,

$$$\frac{1}{\sqrt{1 - u ^{2}}} = \frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}$$$

Gunakan identitas $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

$$$\frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}=\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}}$$$

Dengan asumsi bahwa $$$\cos{\left( v \right)} \ge 0$$$, diperoleh sebagai berikut:

$$$\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}} = \frac{1}{\cos{\left( v \right)}}$$$

Oleh karena itu,

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

Terapkan aturan konstanta $$$\int c\, dv = c v$$$ dengan $$$c=1$$$:

$$- {\color{red}{\int{1 d v}}} = - {\color{red}{v}}$$

Ingat bahwa $$$v=\operatorname{asin}{\left(u \right)}$$$:

$$- {\color{red}{v}} = - {\color{red}{\operatorname{asin}{\left(u \right)}}}$$

Ingat bahwa $$$u=\frac{1}{x}$$$:

$$- \operatorname{asin}{\left({\color{red}{u}} \right)} = - \operatorname{asin}{\left({\color{red}{\frac{1}{x}}} \right)}$$

Oleh karena itu,

$$\int{\frac{1}{x \sqrt{x^{2} - 1}} d x} = - \operatorname{asin}{\left(\frac{1}{x} \right)}$$

Tambahkan konstanta integrasi:

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

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