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5.7E: Ejercicios para la Sección 5.7
https://espanol.libretexts.org/Matematicas/Libro%3A_Calculo_(OpenStax)/05%3A_Integraci%C3%B3n/5.07%3A_Integrales_que_dan_como_resultado_funciones_trigonom%C3%A9tricas_inversas/5.7E%3A_Ejercicios_para_la_Secci%C3%B3n_5.7
Usando la pista, uno tiene $\displaystyle ∫\frac{\csc^2x}{\csc^2x+\cot^2x}\,dx=∫\frac{\csc^2x}{1+2\cot^2x}\,dx.$ Set $u=\sqrt{2}\cot x.$ Then, $du=−\sqrt{2}\csc^2x$ y la integral es\(\displaystyle ...Usando la pista, uno tiene $\displaystyle ∫\frac{\csc^2x}{\csc^2x+\cot^2x}\,dx=∫\frac{\csc^2x}{1+2\cot^2x}\,dx.$ Set $u=\sqrt{2}\cot x.$ Then, $du=−\sqrt{2}\csc^2x$ y la integral es $\displaystyle −\tfrac{1}{\sqrt{2}}∫\frac{du}{1+u^2}=−\tfrac{\sqrt{2}}{2}\tan^{−1}u+C=\tfrac{\sqrt{2}}{2}\tan^{−1}(\sqrt{2}\cot x)+C$ .

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