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

Temukan $$$\int \left(- \frac{1}{\sqrt{a^{2} - x^{2}}}\right)\, dx$$$.

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

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

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

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

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

Jadi,

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

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

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

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

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

Dengan demikian,

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

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

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

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

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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