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Question
`int xcos^3x "d"x`
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Solution
Let I = `int xcos^3x "d"x`
cos3x = 4cos3x − 3cosx
∴ 4cos3x = 3cos x + cos 3x
∴ cos3x = `1/4 (3cos x + cos 3x)`
∴ I = `1/4 int x (3cos x + cos 3x) "d"x`
= `1/4[x int (3cosx + cos3x) "d"x - int{"d"/("d"x)(x) int(3cos x + cos 3x)"d"x}"d"x]`
= `1/4[x(3sinx + (sin3x)/3) - int 1(3sinx + (sin3x)/3)"d"x]`
= `1/4[3x sinx + x/3 sin 3x - (-3 cosx - 1/3 * (cos3x)/3)] + "c"`
∴ I = `1/4(3x sinx + x/3 sin 3x + 3 cos x + 1/9 cos 3x) + "c"`
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