Show : `x^(2)"log"(1)/(x)` is maximum at `x=(1)/(sqrt(e))`

int (log(x+1))/(sqrt(x+1))dx

The solution of the inequality (log)_(1//2) sin^(-1)x >(log)_(1//2)cos^(-1)x is (a) x in [0,(1)/(sqrt(2))] (b) x in [1/(sqrt(2)),1] (c) x in (0,(1)/(sqrt(2))) (d) none of these

Find the derivatives w.r.t. x : sqrt(x^(2)+1)-log[(1)/(x)+sqrt(1+(1)/(x^(2)))]

If f(x)=(tan(pi/4+(log)_e x))^((log)_x e) is to be made continuous at x=1, then what is the value of f(1)?

The greatest and least values of f(x)=tan^(-1)x -(1)/(2) ln x on [(1)/(sqrt(3)),sqrt(3)] are

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

If y=(1)/(3)"log" (x+1)/(sqrt(x^(2)-x+1))+(1)/(sqrt(3))"tan"^(-1)(2x-1)/(sqrt(3)) , show that, (dy)/(dx)=(1)/(x^(3)+1)

Which of the following functions is (are) even, odd, or neither ? (i) f(x)=x^(2)sinx " (ii) " f(x)=sqrt(1+x+x^(2))-sqrt(1-x+x^(2)) (iii) f(x)=log((1-x)/(1+x)) " (iv) " f(x)=log(x+sqrt(1+x^(2))) (v) f(x)=sinx-cosx " (vi) " f(x)=(e^(x)+e^(-x))/(2)

int (1+x^(2))/(1-x^(2)).(dx)/(sqrt(1+x^(4)))=(1)/(sqrt(2))log|(sqrt(1+x^(4))+xsqrt(2))/(1-x^(2))|+c

Show that : lim_(xto0)[(1)/(x)-(log(1+x))/(x^(2))]=(1)/(2)