Questions: Si deseas calcular la integral indefinida ∫ x/(√(1-x^2)) dx, ¿Cuál es la sustitución trigonométrica y la identidad trigonométrica que puedes utilizar para resolverla? u=sec(x), √(1-x^2)=tan(u) u=tan(x), √(1-x^2)=sec(u) u=sin(x), √(1-x^2)=cos(u) u=cos(x), √(1-x^2)=sin(u)

To solve the integral \(\int \frac{x}{\sqrt{1-x^{2}}} dx\), we can use a trigonometric substitution. The appropriate substitution for this integral is \(u = \sin(x)\), because it simplifies the square root term \(\sqrt{1-x^2}\) to \(\cos(u)\). This substitution leverages the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\).

1. Use the substitution \(x = \sin(u)\), which implies \(dx = \cos(u) du\).
2. Substitute \(x\) and \(dx\) in the integral.
3. Simplify the integral using the trigonometric identity \(\sqrt{1 - \sin^2(u)} = \cos(u)\).
4. Integrate with respect to \(u\).
5. Substitute back \(u = \arcsin(x)\) to get the final answer in terms of \(x\).

Para resolver la integral \(\int \frac{x}{\sqrt{1-x^{2}}} dx\), utilizamos la sustitución \(x = \sin(u)\), lo que implica que \(dx = \cos(u) du\).

Sustituimos \(x\) y \(dx\) en la integral: \[ \int \frac{\sin(u)}{\sqrt{1-\sin^2(u)}} \cos(u) du \]

Utilizamos la identidad trigonométrica \(\sqrt{1-\sin^2(u)} = \cos(u)\) para simplificar la integral: \[ \int \frac{\sin(u)}{\cos(u)} \cos(u) du = \int \sin(u) du \]

Integramos con respecto a \(u\): \[ \int \sin(u) du = -\cos(u) + C \]

Sustituimos de vuelta \(u = \arcsin(x)\): \[ -\cos(\arcsin(x)) + C \]

Utilizamos la identidad \(\cos(\arcsin(x)) = \sqrt{1-x^2}\) para obtener la respuesta final: \[ -\sqrt{1-x^2} + C \]

\[ \boxed{-\sqrt{1-x^2} + C} \]