Evaluate: `int(sin^6\ x+cos^6\ x)/(sin^2\ x\ cos^2\ x)`

Explanation

Solution:

As we know, Let I=`\int \frac{\sin ^{6} x+\cos ^{6} x}{\sin ^{2} x \cos ^{2} x} d x`

=`\int \frac{(\sin ^{2} x)^{3}+(\cos ^{2} x)^{3}}{\sin ^{2} x \cos ^{2} x} d x`
=`\int \frac{(\sin ^{2} x+\cos ^{2} x)(\sin ^{4} x+\cos ^{4} x-\sin ^{2} x \cos ^{2} x)}{\sin ^{2} x \cos ^{2} x} dx`
=`\int \frac{(\sin ^{2} x+\cos ^{2} x)^{2}-3 \sin ^{2} x \cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x`

=`\int(\frac{1}{\sin ^{2} x \cos ^{2} x}-3) d x`

=`\int \frac{\sin ^{2}+\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x-3 x+C`

=`\int \sec ^{2} x d x+\int {cosec}^{2} x d x-3 x+C`

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