`x+1/x=3,f i n d t h e v a l u e o fx^4+1/(x^4)`

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 int1/(x+x^5)dx=f(x)+c ,t h e ne v a l u a t eint(x^4)/(x+x^5)dxdot

If int1/(x+x^5)dx=f(x)+c ,t h e ne v a l u a t eint(x^4)/(x+x^5)dxdot

IfI_n=int_0^pix^nsinx dx ,t h e n fin d t h e v a l u eof I_5+20 I_3dot

IfI_n=int_0^pix^nsinx dx ,t h e n fin d t h e v a l u eof I_5+20 I_3dot

: 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: Rvec[-1,oo)a n df(x)=1n([|s in2x|+|cos2x|]) (where [.] is the greatest integer function.) Then, f(x)h a sr a ngZ f(x)i sp e r iod i cw i t hfu n d a m e n t a lp e r iodpi/4 f(x)i sin v e r t i b l ein[0,pi/4] f(x)i sin tofu n c t iondot

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 _________