# Integrate cos(lnx)dx

by Cudi1
Tags: coslnxdx, integrate
 P: 100 1. The problem statement, all variables and given/known data let u=lnx du=1/x*dx dv=cosdx v=-sin 2. Relevant equations Now im confused as im getting nowhere with this substiution, i learned the LIPTE rule but its quite confusing, i have a function within a function 3. The attempt at a solution
 P: 100 for integration by parts to work, i would need two differentiable functions, but the cosdx is not differentiable would it need to be cosxdx for it to be differentiated?
 P: 3,014 Is the integral: $$\int{\cos{(\ln{(x)})} \, dx}$$ If it is, make the substitution: $$t = \ln{(x)} \Rightarrow x = e^{t}$$ and substitute everywhere. The integral that you will get can be integrated by using integration by parts twice, or, if you know complex numbers, by representing the trigonometric function through the complex exponential.
 P: 100 Integrate cos(lnx)dx it is just cos(lnx)dx
 P: 3,014 then proceed as I told you.
 P: 100 ok im getting an integral of the form : coste^tdt. is this correct?
 P: 3,014 yes. proceed by integration by parts or using complex exponentials.
 P: 100 ok thank you for the help, quick question why do we have to let t=lnx?
 P: 3,014 We have a composite function $\cos{(\ln{x})}$ of two elementary functions (trigonometric and logarithmic). As a combination, it does not have an immediate table integral. But, the method of substitution, which is nothing but inverting the chain rule for derivatives of composite functions, works exactly for such compound functions.
 P: 100 thank you, lastly i end up with 1/2(e^tcost-e^tsint), do i sub back , so that x=e^t and t=lnx which leaves me with : 1/2x(cos(lnx)-sin(lnx)+c?
 P: 3,014 I think the sign in front of the sine should be + and the result should be: $$\frac{1}{2} x \left[\cos{\left(\ln{(x)}\right)} + \sin{\left(\ln{(x)}\right)}\right]+ C$$
 P: 100 yes, made a slight mistake thank you very much for the help, i tried doing it another way by letting u=cos(lnx) and dv=dx and i arrived at the same answer
 P: 3,014 yes, the x's will cancel.

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