Question

$\mathrm{If}\int \frac{\sin \left(\ln \left(x\right)\right)}{x}dx=f\left(x\right),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}-f\left(1\right)\mathrm{is}\left(\mathrm{take}\mathrm{the}\mathrm{constant}\mathrm{of}\mathrm{integration}\mathrm{as}0\right)$
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Answer 1

We know the integral of $\sin x$ and $\ln x$ but not of $\sin \left(\ln x\right)$. Also we can see the derivative of $\ln x$ in the integrand.
So we assume $t=\mathrm{lnx}$
$\Rightarrow \mathrm{dt}=\frac{\mathrm{dx}}{x}$
Thus, our integral becomes
$\int \sin t\mathrm{dt}=-\cos \left(t\right)+C$
Also we know that the constant of integration is taken as $0$
Thus substituting back $t$ and taking $C$ as $0$.
We get $f\left(x\right)=-\cos \left(\mathrm{lnx}\right)$
$\Rightarrow -f\left(1\right)=\cos \left(\ln 1\right)=\cos 0=1$