If the third term in the binomial expansion of $${\left(1+x^{{\log }_{2}x}\right)}^5$$ equals 2560, then a possible value of x is -

The positive value of $$\lambda$$ for which the co-efficient of x2 in the expression x2 {\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}} is 720, is -

The value of r for which 20Cr 20C0 + 20Cr$$-$$1 20C1 + 20Cr$$-$$2 20C2 + . . . . .+ 20C0 20Cr is maximum, is

The sum of the real values of x for which the middle term in the binomial expansion of {\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8} equals 5670 is :

Let (x + 10)50 + (x $$-$$ 10)50 = a0 + a1x + a2x2 + . . . . + a50x50, for all x $$\in$$ R; then {{{a_2}} \over {{a_0}}} is equal to

Let Sn = 1 + q + q2 + . . . . . + qn and Tn = 1 + \left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2} + . . . . . .+ {\left( {{{q + 1} \over 2}} \right)^n} where q is a real number and q $$\ne$$ 1. If 101C1 + 101C2 . S1 + .... + 101C101 . S100 = $$\alpha$$T100 then $$\alpha$$ is equal to

The total number of irrational terms in the binomial expansion of (71/5 – 31/10)60 is :

For x2 $$\ne$$ n$$\pi$$ + 1, n $$\in$$ N (the set of natural numbers), the integral \int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} is equal to : (where c is a constant of integration)

Let a \in \left( {0,{\pi \over 2}} \right) be fixed. If the integral \int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx = A(x) cos 2$$\alpha$$ + B(x) sin 2$$\alpha$$ + C, where C is a constant of integration, then the functions A(x) and B(x) are respectively :

The integral \int {{{2{x^3} - 1} \over {{x^4} + x}}} dx is equal to : (Here C is a constant of integration)

If $$\int x^5{e}^{-x^2}dx=g\left(x\right)e^{-x^2}+c$$, where c is a constant of integration, then $$g$$(–1) is equal to :

If \int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C where C is a constant of integration then :

$$\int e^{\sec x}$$ $$(\sec x\tan xf(x)+\sec x\tan x+se{x}^{2}x)dx$$ = esecxf(x) + C then a possible choice of f(x) is :-

The integral $$\int \mathrm{se}{\mathrm{c}}^{2/3}\mathrm{x}\mathrm{cose}{\mathrm{c}}^{4/3}\mathrm{x}\mathrm{dx}$$ is equal to (Hence C is a constant of integration)

If \int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C} where C is a constant of integration, then the function ƒ(x) is equal to

\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx} is equal to (where c is a constant of integration)

The integral \int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx is equal to : (where C is a constant of integration)

The integral $$\int$$cos(loge x) dx is equal to : (where C is a constant of integration)

If \int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx = f(x) $$\sqrt{2x-1}$$ + C, where C is a constant of integration, then f(x) is equal to :

If \int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}} dx = A(x)$${\left(\sqrt{1-x^2}\right)}^m$$ + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))m equals :