Integration of cosx^logx

Question

This is a question of integration and fails to solve this so gv me solution 

 

Answer 1

I don't think that the integral of cosx^lnx  (I assume that that's what you mean) can be expressed by using standard mathematical functions).  At least, that's what Wolfram Alpha says. 

 

The function can be differentiated by using logarithmic differentiation as follows:

 

 y = cosx^lnx

 

lny = ln(cosx^lnx)

 

lny = (lnx)(ln(cosx))

 

y'/y = (1/x)(ln(cosx)) + (lnx)(-sinx/cosx)

 

y' = y[(1/x)ln(cosx) - (lnx)(tanx)]

 

y' = (cosx)^lnx [(1/x)ln(cosx) - (lnx)(tanx)]