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Question
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
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Solution
`int (sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
= `int[((sin^2θ + cos^2θ)^3 - 3sin^2θ*cos^2θ(sin^2θ + cos^2θ))/(sin^2θ*cos^2θ)]*dθ` ...[∵ a3 + b3 = (a + b)3 – 3ab(a + b)]
= `int[((1)^3 - 3sin^2θ*cos^2θ(1))/(sin^2θ*cos^2θ)]*dθ`
= `int[(1)/(sin^2θ*cos^2θ) - 3]*dθ`
= `int [(sin^2θ + cos^2θ)/(sin^2θ*cos^2θ) - 3]*dθ`
= `int (1/cos^2θ + 1/sin^2θ - 3)*dθ`
= `int (sec^2θ + "cosec"^2θ - 3)*dθ`
= `int sec^2θ*dθ + int "cosec"^2θ*dθ - 3int1*dθ`
= tan θ – cot θ - 3θ + c.
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