If `x=2+a+a^2+oo,w h e r e|a|<1a n dy=1+b+b^2+oo,w h e r e|b|<1.` prove that: `1+a b+a^2b^2+oo=(x y)/(x+y-1)`

If x=1+a+a^2+oo,w h e r e|a|<1a n dy=1+b+b^2+oo,w h e r e|b|<1. prove that: 1+a b+a^2b^2+oo=(x y)/(x+y-1)

Prove that t a n(i(log)_e((a-i b)/(a+i b)))=(2a b)/(a^2-b^2)(w h e r ea ,b in R^+)

Find the inverse of the function: f:[1, oo) rarr [1,oo),w h e r ef(x)=2^(x(x-2))

If x^2+y^2=R^2(w h e r eR >0)a n dk=(y prime prime)/((1+yprime^2)^3) then find k in terms of R alone.

If x=a+b i is a complex number such that x^2=3+4i and x^3=2+1i ,w h e r e i=sqrt(-1),t h e n(a+b) equal to ______.

Evaluate the following integral: int_2^4f(x)dx ,\ w h e r e\f(x)={7x+3,\ if\ 1lt=xlt=3 , w h e r e\f(x)=8x \ if\ 3lt=xlt=4

The domain of f(x)=1n(a x^3+(a+b)x^2+(b+c)x+c), where a >0,b^2-4a c=0,i s(w h e r e[dot] represetns greatest integer function). (-1,oo)~(-b/(2a)) (1, oo)~{-b/(2a)} (-1,1)~{-b/(2a)} non eoft h e s e

Find the domain and range of f(x)=log{x},w h e r e{} represents the fractional part function).

The integral int_0^(1. 5)[x^2]dx ,w h e r e[dot] denotoes the greatest integer function, equals ...........

Evaluate the following integrals: int_-5^0f(x)dx ,w h e r ef(x)=|x|+|x+2|+|x+5|