if `3x=cosectheta` and `3/x=cottheta,` f i n d t h e v a l u e o f `3(x^2-1/(x^2))`

If 3x=cosectheta and (3)/(x)=cottheta , find the value of 3(x^(2)-(1)/(x^(2))) .

I fx+1/x=2,t h e p r i n c i p a l v a l u e o f sin^(-1)x is

f(x)=int_1^x(tan^(-1)(t))/t dtAAx in R^+,t h e nfin dt h ev a l u eof f(e^2)-f(1/(e^2))

IfI_n=int_x^pix^nsinx dx ,t h e n fin d t h e v a l u eof I_5+20 I_3dot

If f(0)=1,f(2)=3,f(2)=5,t h e nfin dt h ev a l u eof int_0^1xf"(2x)dx

: If f(x)={x^2+2,xgeq2 1-x ,x 1, 3-x , xlt=0,t h e nt h ev a l u eoflim_(x->1)f(g(x))i s__

f(x)=e^x-e^(-x)-2sinx-2/3x^3dot Then the least value of n for which (d^n)/(dx^n)f(x)(""|)_(x=0)i snon z e roi s (a) 5 (b) 6 (c) 7 (d) 8

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is (a) ( b ) (c) y=( d ) e^(( e ) (f) K(( g ) (h) x-1( i ))( j ))( k ) (l) (m) (b) ( n ) (o) y=( p ) e^(( q ) (r) K e (s))( t ) (u) (v) (c) ( d ) (e) y=( f ) e^(( g ) (h) K(( i ) (j) x-2( k ))( l ))( m ) (n) (o) (d) None of these

Iff(x)=x+int_0^1t(x+t)f(t)dt ,t h e nt h ev a l u eof(23)/2f(0) is equal to _________