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Q17  Find the integrals of the functions \frac{\sin ^ 3x + \cos ^ 3x }{\sin ^ 2 x \cos ^2 x }

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best_answer

\frac{\sin ^ 3x + \cos ^ 3x }{\sin ^ 2 x \cos ^2 x } 

now splitting the terms we can write

\\=\frac{\sin^3x}{\sin^2x.\cos^2x}+\frac{\cos^3x}{\sin^2x.\cos^2x}\\ =\frac{\sin x}{cos^2x}+\frac{\cos x}{\sin^2x}\\ =\tan x.\sec x+\cot xcosec x

Therefore, the integration of 
                                               \frac{\sin ^ 3x + \cos ^ 3x }{\sin ^ 2 x \cos ^2 x } 

                                             \\=\int (\tan x\sec x+\cot xcosec x)dx\\ =\sec x-cosec\ x+C

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manish

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