Question

$IfF\left(x\right)=\int e^x\left(\sin \left(\mathrm{lnx}\right)+\frac{\cos \left(\mathrm{lnx}\right)}{x}\right]\mathrm{dx},\mathrm{then}F\left(1\right)\left(\mathrm{assume}\mathrm{the}\mathrm{constant}\mathrm{of}\mathrm{integration}\mathrm{to}\mathrm{be}0\right)\mathrm{is}$

Answer 1

For solving these types of integral which has $e^x$ multiplied with some other functions in the integrand we should check whether the function is of the form $\int e^x\left(f\left(x\right)+f^{\prime }\left(x\right)\right]dx$
because
$\int e^x\left(f\left(x\right)+f^{\prime }\left(x\right)\right]dx=e^{x}f\left(x\right)+C.$
Now Looking at this integral we see that if $f\left(x\right)=\sin \left(\ln x\right)$,
$\Rightarrow f^{\prime }\left(x\right)$$=\frac{\cos \left(\ln x\right)}{x}$,
thus our integral reduces to the form $\int e^x\left(f\left(x\right)+f^{\prime }\left(x\right)\right]dx$ $\Rightarrow \int e^x\left(\sin \left(\ln x\right)+\frac{\cos \left(\ln x\right)}{x}\right]dx$
$=e^x\sin \left(\ln x\right)+C$
Now, the constant of integration is assunmed to be zero.
$\Rightarrow F\left(x\right)=e^x\sin \left(\ln x\right)$
$F\left(1\right)=e^1\sin \left(\ln 1\right)$$\Rightarrow F\left(1\right)=e\times 0=0$