Evaluate `int_(-1)^(3)(x+[x])dx` ,where [.] denotes the greatest integer function.

Answer

`a rarrp; b rarr s; c rarr r; d rarrp`

Explanation

The locus of point P satisfying `PA-PB=2` is a branch of the hyperbola `x^(2)-y^(2)//3=1`.
For r = 2, the circle and the branch of the hyperbola intersect at two points. For r = 1, there is one point of intersection.
If m is the slope of the common tangent, then
`m^(2)-3=r^(2)(1+m^(2))`
`"or "x^(2)=(r^(2)+3)/(1-r^(2))`
Hence, there are no common tangents for r gt 1 and two common tangents for r lt 1.