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Question
Evaluate the following : `int x.cos^3x.dx`
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Solution
cos 3x = 4 cos3x – 3cos x
∴ cos 3x + 3 cos x = 4 cos3x
∴ `int cos^3x = (1)/(4) cos3x + (3)/(4) cosx`
∴ `int cos^3x.dx = (1)/(4) int cos3x.dx + (3)/(4) int cos x.dx`
= `(1)/(4)((sin3x)/3) + (3)/(4) sinx`
= `(sin3x)/(12) + (3sinx)/(4)` ...(1)
Let I = `int x cos^3x.dx`
= `x int cos^3x.dx - int[{d/dx (x) int cos^3x.dx}].dx`
= `x[(sin3x)/(12) + (3sinx)/(4)]- int 1.((sin3x)/(12) + (3sinx)/4).dx` ...[By (1)]
= `(xsin3x)/(12) + (3x sinx)/(4) - (1)/(12) int sin 3x.dx - 3/4 int sin x.dx`
= `(x sin3x)/(12) + (3xsinx)/(4) - (1)/(12) ((-cos3x)/3) - (3)/(4) (- cos x) + c`
= `(1)/(4)[x/3 sin 3x + 1/9 cos3x + 3x sin x + 3 cos x] + c`.
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