Question

If the integral of the function $\frac{sin\left(lnx\right)}{x}=f\left(x\right)$, then find the value of f(1)
(take constant of integration equal to zero)
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Answer 1

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
$\Rightarrow dt=\frac{dx}{x}Soweget\int \frac{sin\left(lnx\right)}{x}dx=\int sin\left(t\right)dt=-cost$
Replacing t with lnx, we get
f(x) = - cos (lnx)
We want to find f(1)
f(1) = - cos(ln(1)
= - cos(0)
= -1