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

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

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

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

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

Por lo tanto,

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

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

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

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

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

La integral se convierte en

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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