Question

If the integral $I=\int \left(- \frac{sin x}{x} - ln x cos x\right)dx$ $=f\left(x\right)+C$ (where, $C$ is the constant of integration) and $f\left(e\right)=-sine,$ then the number of natural numbers less than $\left[f \left(\frac{\pi }{6}\right)\right]$ is equal to (where, $\left[\cdot \right]$ is the greatest integer function)

Answer 1

Given Integral is $I=-\int d\left(sin x \cdot ln x\right)$
$=-sinx\cdot lnx+C$
$\therefore f\left(x\right)=-sinx\cdot lnx$
$\Rightarrow f\left(\frac{\pi }{6}\right)=-\frac{1}{2}ln\left(\frac{\pi }{6}\right)=\frac{1}{2}ln\left(\frac{6}{\pi }\right)$
$\Rightarrow f\left(\frac{\pi }{6}\right)\in \left(0 , 1\right)\Rightarrow \left[f \left(\frac{\pi }{6}\right)\right]=0$