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Question
Evaluate the following : `int x.sin^2x.dx`
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Solution
`int x.sin^2x.dx`
= `int x((1 - cos2x)/2).dx`
= `(1)/(2) int (x - x cos2x).dx`
= `(1)/(2) int x.dx - (1)/(2) int x cos 2x.dx`
= `(1)/(2).x^2/(2) - (1)/(2)[x int cos 2x.dx - int {d/dx (x) int cos 2x.dx}.dx]`
= `x^2/(4) - (1)/(2)[x. (sin2x)/(2) - int 1. (sin2x)/(2).dx]`
= `x^2/(4) - (1)/(2) x. sin2x + (1)/(4) sin 2x.dx`
= `x^2/(4) - (1)/(4) x.sin2x + (1)/(4).((-cos2x))/(2) + c`
= `x^2/(4) - (1)/(4) x.sin2x - (1)/(8) cos 2x + c`
= `(1)/(4) [x^2 - x.sin 2x - (1)/(2) cos 2x] + c`.
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