What is the integral int x*cos 6x dx

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int x*cos 6x dx` has to be determined.

This can be done using a method called integration by parts. The integral `int u dv = u*v - int v du`

For `int x*cos 6x dx` , let `u = x` and `dv = cos 6x dx`

If u = x, du = dx and `v = int cos 6x dx = (1/6)*sin 6x`

`int x*cos 6x dx = x*(1/6)*sin 6x - int (1/6)*sin 6x dx`

= `(x*sin 6x)/6 - (1/6)*(1/6)*(-cos 6x)`

= `(x*sin 6x)/6 + (cos 6x)/36 + C`

The integral `int x*cos 6x dx = (x*sin 6x)/6 + (cos 6x)/36 + C`