Question

ejercicios de integrales inmediatas
∫Tan(x)2*Csc(x)Sec(x)2 dx

Answer 1

PREGUNTA

                             Resolver ∫Tan(x)2*Csc(x)Sec(x)2 dx

SOLUCIÓN

Hola!! :D

Resolviendo la integral

                               $\int\limits {\tan^2(x)*\csc(x)*\sec^2(x)} \, dx \\\\Expresamos \: en \: senos \: y \:cosenos\\\\\int\limits {\dfrac{\sin^2(x)}{\cos^2(x)}*\dfrac{1}{\sin(x)}*\dfrac{1}{\cos^2(x)}} \, dx\\\\\\\int\limits {\dfrac{\sin(x)}{\cos^4(x)}} \, dx, usamos \tan(x) = \dfrac{\sin(x)}{\cos(x)} \\\\\\\int\limits {\tan(x)*\sec^3(x)} \, dx\\\\Sustituci\'on\\\\u = \sec^3{(x)} \Rightarrow du = 3\sec^3(x)\tan(x)dx\\\\\Rightarrow dx = \dfrac{du}{3\sec^3(x)\tan(x)}$

                               $\int\limits {\tan(x)*\sec^3(x)} \, dx\\\\\\\int\limits {\tan(x) *u} \,(\dfrac{du}{3\sec^3(x)\tan(x)})\\\\\\\int\limits {\tan(x) *u} \,(\dfrac{du}{3(u)\tan(x)})\\\\\\\int\limits\dfrac{1}{3} \, du = \dfrac{1}{3}u + C = \boxed{\boxed{\boldsymbol{\dfrac{1}{3}\sec^3(x) + C}}}$