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Question

If the integral of the function sin(lnx)x=f(x), then find the value of f(1)
(take constant of integration equal to zero)
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Solution

We know the integral of sinx and lnx separately but not of the function sin(lnx). But we have the derivative of lnx in the given function. We will replace lnx with t and proceed
t=ln x
dt=dxxSo we get sin(lnx)xdx=sin(t)dt=cost
Replacing t with lnx, we get
f(x) = - cos (lnx)
We want to find f(1)
f(1) = - cos(ln(1))
= - cos(0)
= -1

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