The integration of $$\int {{\mathop{\rm logx}\nolimits} \,dx} $$ has the value

The volume generated by revolving the area bounded by the parabola $${y^2} = 8x$$ and the line $$x=2$$ about $$y$$-axis is

The directional derivative of $$f\left( {x,y} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$ at point $$P\left( {2,1,3} \right)\,\,$$ in the direction of the vector $$\,\,a = \overrightarrow i - 2\overrightarrow k \,\,$$ is

A switched mode power supply operating at $$20$$ $$kHz$$ to $$100$$ $$kHz$$ range uses as the main switching element