For all `x in (0,1)` (a) `e^x<1+x` (b) `(log)_e (1+x) < x` (c) `sin x > x` (d) `(log)_e x > x`

lim_(x->oo)n^2(x^(1/n)-x^(1/((n+1)))),x >0 , is equal to (a)0 (b) e^x (c) (log)_e x (d) none of these

The differential coefficient of f((log)_e x) with respect to x , where f(x)=(log)_e x , is (a) x/((log)_e x) (b) 1/x(log)_e x (c) 1/(x(log)_e x) (d) none of these

f(x)=|x log_e x| monotonically decreases in (0,1/e) (b) (1/e ,1) (1,oo) (d) (1/e ,oo)

The primitive of the function f(x)=(1-1/(x^2))a^(x+1/x)\ ,\ a >0 is (a) (a^(x+1/x))/((log)_e a) (b) (log)_e adota^(x+1/x) (c) (a^(x+1/x))/x(log)_e a (d) x(a^(x+1/x))/((log)_e a)

Integration of 1/(1+((log)_e x)^2) with respect to (log)_e x is (tan^(-1)((log)_e x)/x)+C (b) tan^(-1)((log)_e x)+C (c) (tan^(-1)x)/x+C (d) none of these

Differentiate the following w.r.t. x:(i) e^(-x) (ii) sin (log x), x > 0 (iii) cos^(-1)(e^x) (iv) e^(cosx)

int_(1//e)^(e) (dx)/(x(log x)^(1//3))

Find the range of f(x)=(log)_e x-((log)_e x)^2/(|(log)_e x|)

The differentiation of (log)_a x (a >0) with respect to x i.e. d/(dx)((log)_a x)= 1/(x(log)_e a)

If y=tan^(-1){((log)_e(e//x^2))/((log)_e(e x^2))}+tan^(-1)((3+2\ (log)_e x)/(1-6\ (log)_e x)) , then (d^2y)/(dx^2)= (a) 2 (b) 1 (c) 0 (d) -1