If `x<0,t h e ntan^(-1)x` is equal to

If -1 lt x lt 0 , then cos^(-1) x is equal to

If f(x)=(log)_x(log x),t h e nf^(prime)(x) at x=e is equal to (a) 1/e (b) e (c) 1 (d) zero

If int_1^2e^(x^2)dx=a ,t h e nint_e^(e^4)sqrt(1n x)dx is equal to (a) 2e^4-2e-a (b) 2e^4-e-a (c) 2e^4-e-2a (d) e^4-e-a

If f(x)=sin^(-1)((sqrt(3))/2x-1/2sqrt(1-x^2)),-1/2lt=xlt=1,t h e nf(x) is equal to

If f (x) ={{:(x",", "when " 0 lt x lt 1),(2-x",", "when" 1 lt x le 2):} then f'(1) is equal to-

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

If f(x)=cosx-int_0^x(x-t)f(t)dt ,t h e nf^(primeprime)(x)+f(x) is equal to (a) -cosx (b) -sinx (c) int_0^x(x-t)f(t)dt (d) 0

If y=cos^(-1)sqrt((sqrt(1+x^2)+1)/(2sqrt(1+x^2))),t h e n(dy)/(dx) is equal to (a) 1/(2(1+x^2)),x in R (b) 1/(2(1+x^2)),x >0 (c) (-1)/(2(1+x^2)),x<0 (d) 1/(2(1+x^2)),x<0

If lambda=int_0^1(e^t)/(1+t) , then int_0^1e^tlog_e(1+t)dt is equal to

Column I, a) int(e^(2x)-1)/(e^(2x)+1)dx is equal to b) int1/((e^x+e^(-x))^2)dx is equal to c) int(e^(-x))/(1+e^x)dx is equal to d) int1/(sqrt(1-e^(2x)))dx is equal to COLUMN II p) x-log[1+sqrt(1-e^(2x)]+c q) log(e^x+1)-x-e^(-x)+c r) log(e^(2x)+1)-x+c s) -1/(2(e^(2x)+1))+c