this problem is under the integration by parts section in my book ... but im stuck and really don't know where to start.
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This one is hard to explain over a computer, however, I will try.
It involves substituting prior to integration by parts.
Start by letting w = ln(6x), so dw = (1/x) dx.... xdw = dx
The x in xdw is a problem. We go back to w = ln(6x)
rewrite as w = log (base e) 6x...
The w is now the exponent that goes on the e to equal 6x
e^w = 6x
(1/6) e^w = x
instead of xdw = dx.... (1/6) (e^w)dw = dx
your integral becomes (1/6) (e^w)cos(w)dw
let u = e^w and dv = cos(w) dw
Go to Work!
It involves substituting prior to integration by parts.
Start by letting w = ln(6x), so dw = (1/x) dx.... xdw = dx
The x in xdw is a problem. We go back to w = ln(6x)
rewrite as w = log (base e) 6x...
The w is now the exponent that goes on the e to equal 6x
e^w = 6x
(1/6) e^w = x
instead of xdw = dx.... (1/6) (e^w)dw = dx
your integral becomes (1/6) (e^w)cos(w)dw
let u = e^w and dv = cos(w) dw
Go to Work!
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w = ln 6x ---> dw = dx / x...but x = (1/6) e^w---> (1/6) e^w cos w dw..now IBP