Find \int \frac{\sin ^3x+\cos ^3x}{\sin ^2x \cos ^2x}dx

 

 

 

 

 
 
 
 
 

Answers (1)

\int \frac{\sin ^3x+\cos ^3x}{\sin ^2x \cos ^2x}dx=I

Let I=\int \frac{\sin ^3x+\cos ^3x}{\sin ^2x \cos ^2x}dx

         =\int \frac{\sin ^3xdx}{\sin ^2x \cos ^2x}+\int \frac{\cos ^3xdx}{\sin ^2x \cos ^2x}

          =\int \tan x\sec xdx+\int \cot x\cdot \csc x dx

 =\sec x-\csc x+c     \left [ \because \int \tan x\cdot \sec xdx=\sec x\: \: and\: \: \int \cot x\cdot \csc xdx=-\csc x \right ]

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