Integral dari $$$\frac{1}{\sqrt{4 x^{2} - 5}}$$$

Temukan $$$\int \frac{1}{\sqrt{4 x^{2} - 5}}\, dx$$$.

Misalkan $$$x=\frac{\sqrt{5} \cosh{\left(u \right)}}{2}$$$.

Maka $$$dx=\left(\frac{\sqrt{5} \cosh{\left(u \right)}}{2}\right)^{\prime }du = \frac{\sqrt{5} \sinh{\left(u \right)}}{2} du$$$ (langkah-langkah dapat dilihat »).

Selain itu, berlaku $$$u=\operatorname{acosh}{\left(\frac{2 \sqrt{5} x}{5} \right)}$$$.

Oleh karena itu,

$$$\frac{1}{\sqrt{4 x^{2} - 5}} = \frac{1}{\sqrt{5 \cosh^{2}{\left( u \right)} - 5}}$$$

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

$$$\frac{1}{\sqrt{5 \cosh^{2}{\left( u \right)} - 5}}=\frac{\sqrt{5}}{5 \sqrt{\cosh^{2}{\left( u \right)} - 1}}=\frac{\sqrt{5}}{5 \sqrt{\sinh^{2}{\left( u \right)}}}$$$

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

$$$\frac{\sqrt{5}}{5 \sqrt{\sinh^{2}{\left( u \right)}}} = \frac{\sqrt{5}}{5 \sinh{\left( u \right)}}$$$

Dengan demikian,

$${\color{red}{\int{\frac{1}{\sqrt{4 x^{2} - 5}} d x}}} = {\color{red}{\int{\frac{1}{2} d u}}}$$

Terapkan aturan konstanta $$$\int c\, du = c u$$$ dengan $$$c=\frac{1}{2}$$$:

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

Ingat bahwa $$$u=\operatorname{acosh}{\left(\frac{2 \sqrt{5} x}{5} \right)}$$$:

$$\frac{{\color{red}{u}}}{2} = \frac{{\color{red}{\operatorname{acosh}{\left(\frac{2 \sqrt{5} x}{5} \right)}}}}{2}$$

Oleh karena itu,

$$\int{\frac{1}{\sqrt{4 x^{2} - 5}} d x} = \frac{\operatorname{acosh}{\left(\frac{2 \sqrt{5} x}{5} \right)}}{2}$$

Tambahkan konstanta integrasi:

$$\int{\frac{1}{\sqrt{4 x^{2} - 5}} d x} = \frac{\operatorname{acosh}{\left(\frac{2 \sqrt{5} x}{5} \right)}}{2}+C$$

$$$\int \frac{1}{\sqrt{4 x^{2} - 5}}\, dx = \frac{\operatorname{acosh}{\left(\frac{2 \sqrt{5} x}{5} \right)}}{2} + C$$$A