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Question

If sin(ln(x))xdx=f(x), then the value of f(1) is (take the constant of integration as 0)
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Solution

We know the integral of sinx and lnx but not of sin(lnx). Also we can see the derivative of lnx in the integrand.
So we assume t=lnx
dt=dxx
Thus, our integral becomes
sin t dt=cos(t)+C
Also we know that the constant of integration is taken as 0
Thus substituting back t and taking C as 0.
We get f(x)=cos(lnx)
f(1)=cos(ln1)=cos0=1

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