Show that `1/(2x+1)+1/(3.(2x+1)^3)+1/(5(2x+1)^5)+....=log_(e)sqrt((x+1)/x)`

2[(1)/(2x+1)+(1)/(3*(2x+1)^(3))+(1)/(5*(2x+1)^(5))+…..oo]=

(1)/(2x-1)+(1)/(3).(1)/((2x-1)^(3))+(1)/(5)(1)/((2x-1)^(5))+....=

(1)/(2x+1)+(1)/(3)(1)/((2x+1)^(3))+(1)/(5)(1)/((2x+1)^(5))+....=

Show that Lt_(x to 0)(log(1+x))/(x^(2))=1

A : (1)/(2)-(1)/(2).(1)/(2^(2))+(1)/(3).(1)/(2^(3))-(1)/(4).(1)/(2^(4))+....=log_(e)((3)/(2)) R : log_(e)(1+x)=x-(x^(2))/(2)+(x^(3))/(3)-(x^(4))/(4)+...

Show that int_(1)^(7thsqrt(2))(1)/(x(2x^(7) + 1))dx=(1)/(7) log((6)/(5))

Show that int_(0)^(1//2) (dx)/((1-2x^(2))sqrt(1-x^(2))) = (1)/(2) log(2+sqrt(3))

Show that int_(1)^(2)(x+(1)/(x))^(3//2)((x^(2)-1)/(x^(2)))dx = ((5)/(2))^(3//2) - (8sqrt(2))/(5)

Show that int_(0)^(1//2)(x sin^(-1)x)/(sqrt(1-x^(2)))dx = (1)/(2)-(sqrt(3))/(12)pi

If |x| lt 1 then (1)/(2) log_(e)((1+x)/(1-x)) =