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Question
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Solution
Let I = `int (1)/(cosx - sinx).dx`
Dividing each term by `sqrt(1^2 + (-1)^2) = sqrt(2)`, we get
I = `(1)/sqrt(2) int (1)/(cosx. 1/sqrt(2) - sinx. 1/sqrt(2)).dx`
= `1/sqrt(2) int (1)/(cosx . cos pi/(4) - sin x. sin pi/(4)).dx`
= `1/sqrt(2) int (1)/(cos(x + pi/4)).dx`
= `1/sqrt(2) int sec(x + pi/4).dx`
= `1/sqrt(2)log|sec(x + pi/4) + tan(x + pi/4)| + c`.
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