" 10."(1)/(sqrt(x^(2)+2x+2))

Rationalise the denominator: (a) (1)/(root(3)(3) + root(3)(2)) , (b) (2)/(sqrt5 + sqrt3 + sqrt2) , (c) (x^(2))/(sqrt(x^(2) + y^(2)) - y) , (d) (1)/(sqrt6 + sqrt5 - sqrt11) (e) (sqrt(x + 2y) - sqrt(x -2y))/(sqrt(x + 2y) + sqrt(x - 2y)) , (f) (sqrt10 + sqrt5 - sqrt3)/(sqrt10 - sqrt5 + sqrt3)

IfI=int(dx)/(x^(3)sqrt(x^(2)-1)), then Iequals a.(1)/(2)((sqrt(x^(2)-1))/(x^(3))+tan^(-1)sqrt(x^(2)-1))+C b.(1)/(2)((sqrt(x^(2)-1))/(x^(2))+x tan^(-1)sqrt(x^(2)-1))+Cc(1)/(2)((sqrt(x^(2)-1))/(x^(2))+tan^(-1)sqrt(x^(2)-1))+Cd(1)/(2)((sqrt(x^(2)-1))/(x^(2))+tan^(-1)sqrt(x^(2)-1))+C

(1)/(sqrt(2x))+sqrt(2x)

If 2x = sqrt(a) - (1)/(sqrt(a)) , then the value of (sqrt(x^(2) + 1))/(x + sqrt(x^(2) +1)) is

If x+sqrt(x^(2)-1)+(1)/(x+sqrt(x^(2)+1))=20 then x^(2)+sqrt(x^(4)-1)+(1)/(x^(2)+sqrt(x^(4)-1))=

lim_(x rarr1)(sqrt(x^(2)+8)-sqrt(10-x^(2)))/(sqrt(x^(2)+3)-sqrt(5-x^(2)))=

(d)/(dx)[cos^(-1)(x sqrt(x)-sqrt((1-x)(1-x^(2))))]=(1)/(sqrt(1-x^(2)))-(1)/(2sqrt(x-x^(2)))(-1)/(sqrt(1-x^(2)))-(1)/(2sqrt(x-x^(2)))(1)/(sqrt(1-x^(2)))+(1)/(2sqrt(x-x^(2)))(1)/(sqrt(1-x^(2)))0 b.1/4c.-1/4d none of these

Simplify : (a) sqrt(y+sqrt(2xy-x^(2))) + sqrt(y-sqrt(2xy-x^(2))) (b) (x+sqrt(x^2-1))/(x-sqrt(x^(2)-1)) -(x-sqrt(x^(2)-1))/(x+sqrt(x^(2)-1))

Find the integral of (1)/(sqrt(x^2-a^2)) with respect to x and hence evaluate (1)/(sqrt(x^2+4x-10))dx .

int(x^(4)-1)/(x^(2)sqrt(x^(2)+x^(2)+1))dx=sqrt(x^(2)+(1)/(x^(2))+1)+C(sqrt(x^(2)+x^(2)+1))/(x^(2))+C(sqrt(x^(4)+x^(2)+1))/(x)+C(d) none of these