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CAL1 • Integrales
CAL1_INT_056
Guía de ejercicios
Enunciado
Evaluar:
$$ \int \tan^{-1}\left(\frac{1 - \sin x}{\cos x}\right) dx $$
$$ \int \tan^{-1}\left(\frac{1 - \sin x}{\cos x}\right) dx $$
Solución Paso a Paso
1. Simplificación:
Similar a casos anteriores:
$1 - \sin x = 2\sin^2(\frac{\pi}{4} - \frac{x}{2})$
$\cos x = \sin(\frac{\pi}{2} - x) = 2\sin(\frac{\pi}{4} - \frac{x}{2})\cos(\frac{\pi}{4} - \frac{x}{2})$
La fracción simplifica a:
$$ \frac{\sin(\frac{\pi}{4} - \frac{x}{2})}{\cos(\frac{\pi}{4} - \frac{x}{2})} = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) $$
2. Integración:
$$ \int \left(\frac{\pi}{4} - \frac{x}{2}\right) dx = \frac{\pi}{4}x - \frac{x^2}{4} + C $$
Resultado:
$$ \boxed{\frac{\pi x - x^2}{4} + C} $$
Similar a casos anteriores:
$1 - \sin x = 2\sin^2(\frac{\pi}{4} - \frac{x}{2})$
$\cos x = \sin(\frac{\pi}{2} - x) = 2\sin(\frac{\pi}{4} - \frac{x}{2})\cos(\frac{\pi}{4} - \frac{x}{2})$
La fracción simplifica a:
$$ \frac{\sin(\frac{\pi}{4} - \frac{x}{2})}{\cos(\frac{\pi}{4} - \frac{x}{2})} = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) $$
2. Integración:
$$ \int \left(\frac{\pi}{4} - \frac{x}{2}\right) dx = \frac{\pi}{4}x - \frac{x^2}{4} + C $$
Resultado:
$$ \boxed{\frac{\pi x - x^2}{4} + C} $$