Integral de $$$\frac{1}{\sqrt{a^{2} - x^{2}}}$$$ con respecto a $$$x$$$

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

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

Entonces $$$dx=\left(\sin{\left(u \right)} \left|{a}\right|\right)^{\prime }du = \cos{\left(u \right)} \left|{a}\right| du$$$ (los pasos pueden verse »).

Además, se sigue que $$$u=\operatorname{asin}{\left(\frac{x}{\left|{a}\right|} \right)}$$$.

Por lo tanto,

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

Utiliza la identidad $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

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

Suponiendo que $$$\cos{\left( u \right)} \ge 0$$$, obtenemos lo siguiente:

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

Por lo tanto,

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

Aplica la regla de la constante $$$\int c\, du = c u$$$ con $$$c=1$$$:

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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