Evaluate,`int_(r1)^(r_2) (kq1q2)/r^2 dr.`

Find the value of sum_(r=1)^(n) (r)/(1+r^(2)+r^(4))

The value of int_(-2)^(2)[p log((1+x)/(1-x))+q log ((1-x)/(1+x))^(-2)+r]dx depends on -

Show that , v = (A)/(r) + B satisfies the differential equation (d^(2) v)/(dr^(2)) + (2)/(r) .(dv)/(dr) = 0

The value of sum_(r=1)^(49)(2r^(2) - 48r +1)/((50-r).""^(50)C_(r)) is "_____" .

Two concentric thin metallic spherical shells of radii R_(1) and R_(2) (where R_(1) lt R_(2) ) are charged with q_(1) and q_(2) coulombs, respectively. Using Gauss' theorem show that, (i) the electric field intensity for radius r lt R_(1) is zero, (ii) the electic field intensity for radius r, where R_(1) lt r lt R_(2), is (1)/(4pi epsilon_(0)).(q_(1))/(r^(2)) and (iii) the electri field intensity for radius r gt R_(2) is (1)/(4pi epsilon_(0)).(q_(1)+q_(2))/(r^(2)) .

Write the series Sigma_(r=1)^(n) (r^(2)+1)/(2r^(2) - 1) in expanded form.

Evaluate underset(nrarrinfty)lim 1/n overset(4n)underset(r=1)sum r/(sqrt(n^2+r^2)

Write the series Sigma_(r=1)^(n) (2r+1)/(r^(2) + 1) in expanded form.

If r=2(1-costheta) , then show that, int_(0)^(pi)sqrt(r^(2)+((dr)/(d theta))^(2))d theta=8

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))