For ` 0 lt x le pi`, Sinh`""^(-1)` (cot x ) =

For 0 lt x le pi, sin h^(-1)(cot x) =

For 0 lt x le pi, sinh^(-1) (cotx)=

If f(x)={:{((sin2x)/(sqrt(1-cos2x))", for " 0 lt x le pi/2),((cos x)/(pi-2x)", for "pi/2 lt x le pi):}

If f(x)={:{(x sin x", for " 0 lt x le pi/2),(pi/2 sin (pi+x)", for " pi/2 lt x lt pi):} , then

If f(x)={:{(x sin x", for " 0 lt x le pi/2),(pi/2 sin (pi+x)", for " pi/2 lt x lt pi):} , then

Find the values of a and b so that the function f(x)={{:(x+asqrt2 sin x", "0 le x le pi//4),(2x cot x + b", "pi//4 le x le pi//2),(a cot 2x - b sin x", "pi//2 lt x le pi):} is continuous for 0 le x le pi .

For 0 lt x le (pi)/(2) , show that x - (x^(3))/(6) lt sin x

For each ositive integer n, define a function f _(n) on [0,1] as follows: f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):} Then the value of lim _(x to oo)int _(0)^(1) f_(n) (x) dx is: