I
CAL1 • Integrales
CAL1_INT_232
Guía de Ejercicios de Cálculo II
Enunciado
Evaluar:
$$ \int \frac{dx}{x\sqrt{x^2 + 4}} $$
$$ \int \frac{dx}{x\sqrt{x^2 + 4}} $$
Solución Paso a Paso
Sustitución $x = \frac{1}{t}$, $dx = -\frac{dt}{t^2}$.
$$ \int \frac{-\frac{dt}{t^2}}{\frac{1}{t} \sqrt{\frac{1}{t^2} + 4}} = -\int \frac{dt}{\sqrt{1 + 4t^2}} $$
Hacemos $2t = \sinh(u)$ o simplemente usamos la fórmula directa:
$$ -\frac{1}{2} \ln\left| 2t + \sqrt{4t^2 + 1} \right| + C = -\frac{1}{2} \ln\left| \frac{2 + \sqrt{x^2+4}}{x} \right| + C $$
$$ \boxed{\frac{1}{2} \ln\left| \frac{x}{2 + \sqrt{x^2+4}} \right| + C} $$
$$ \int \frac{-\frac{dt}{t^2}}{\frac{1}{t} \sqrt{\frac{1}{t^2} + 4}} = -\int \frac{dt}{\sqrt{1 + 4t^2}} $$
Hacemos $2t = \sinh(u)$ o simplemente usamos la fórmula directa:
$$ -\frac{1}{2} \ln\left| 2t + \sqrt{4t^2 + 1} \right| + C = -\frac{1}{2} \ln\left| \frac{2 + \sqrt{x^2+4}}{x} \right| + C $$
$$ \boxed{\frac{1}{2} \ln\left| \frac{x}{2 + \sqrt{x^2+4}} \right| + C} $$