Evaluate the indefinite integral. integrate of cot(x)ln(sin(x))dx
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calendarEducator since 2011
write5,349 answers
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You should evaluate the integral using the following substitution such that:
`sin x = t => cos x dx = dt`
Changing the variable yields:
`int ln(sin x) cos x dx = int ln t dt`
You should use integration by parts formula such that:
`int udv = uv - int vdu`
Reasoning by analogy yields:
`u = ln t => du = (dt)/t`
`dv = dt => v =` `t`
`int ln t dt = t*ln t - int t*(1/t)dt`
`int ln t dt = t*ln t - int dt => int ln t dt = t*ln t - t + c`
Factoring out t yields:
`int ln t dt = t*(ln t - 1) + c`
Substituting back `sin x` for t yields:
`int ln(sin x) cos x dx = sin x*(ln(sin x) - 1) + c`
Hence, evaluating the given indefinite integral , using two methods of integration, yields `int ln(sin x) cos x dx = sin x*(ln(sin x) - 1) + c.`
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calendarEducator since 2012
write1,657 answers
starTop subjects are Math, Science, and Social Sciences
Let;
`t = lnsinx`
Then;
`(dt)/dx = 1/sinx*cosx = cotx`
`dt = cotxdx`
`int cotxlnsinxdx`
`= int lnsinxcotxdx`
`= int tdt`
`= t^2/2+C` where C is a constant
`= 1/2(lnsinx)^2+C`
`= lnsinx+C`
`int cotxlnsinxdx = lnsinx+C`