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

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

Misalkan $$$x=2 \sin{\left(u \right)}$$$.

Maka $$$dx=\left(2 \sin{\left(u \right)}\right)^{\prime }du = 2 \cos{\left(u \right)} du$$$ (langkah-langkah dapat dilihat »).

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

Integran menjadi

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

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

$$$\frac{1}{\sqrt{4 - 4 \sin^{2}{\left( u \right)}}}=\frac{1}{2 \sqrt{1 - \sin^{2}{\left( u \right)}}}=\frac{1}{2 \sqrt{\cos^{2}{\left( u \right)}}}$$$

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

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

Oleh karena itu,

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

Terapkan aturan konstanta $$$\int c\, du = c u$$$ dengan $$$c=1$$$:

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

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

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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