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How do you integrate #int sinxln(cosx)# by integration by parts method?

1 Answer

Aug 11, 2016

# -cos(x)loge(cos(x))+cos(x) + C#

Explanation:

#d/(dx)(cos(x)log_e(cos(x)))=-sin(x) log_e(cos(x))-sin(x)#

so

#int sin(x) log_e(cos(x))dx =#
#= -int d/(dx)(cos(x)log_e(cos(x)))dx-int sin(x) dx = #
#= -cos(x)loge(cos(x))+cos(x) + C#

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