If `S_n=sum_(r=1)^n r !(n >6),"t h e nf i n dt h ev a l u eo f"` `[cot^(-1)(cot(s_n-7[(S_n)/7]))],"w h e r e"[dot]"d e n o t e sg r e a t e s t"` integer function

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

int_0^x(2^t)/(2^([t]))dt ,w h e r e[dot] denotes the greatest integer function and x in R^+ , is equal to

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))

Let R=(5sqrt(5)+11)^(2n+1)a n df=R-[R]w h e r e[] denotes the greatest integer function, prove that Rf=4^(2n+1)

Prove that int_0^oo[n e^(-x)]dx=1n((n^n)/(n !)),w h e r en is a natural number greater than 1 and [.] denotes the greatest integer function..

Prove that int_0^oo[n e^(-x)]dx=ln((n^n)/(n !)),w h e r en is a natural number greater than 1 and [.] denotes the greatest integer function..

If A=[1ta n x-t a n x1],t h e n detA^T A^(-1) is

Each question contains statements given in two columns which have to be matched. Statements a,b,c,d in column I have to be matched with statements p,q,r,s in column II. If the correct match are a-p, a-s, b-r, c-p, c-q, and d-s, then the correctly bubbled 4 x 4 matrix should be as follows: fig Column I, Column II If L=""("lim")_(xvec-1)((7-x)^(1/3)-2)/((x+1)),t h e n12 L= , p. -2 If L=""("lim")_(xvecpi/4)(tan^3x-tanx)/(cos(x+pi/4)),t h e n-L/4= , q. 2 If L=""("lim")_(xvec1)((2x-3)(sqrt(x)-1))/(2x^2+x-3),t h e n20 L= , r. 1 If L=""("lim")_(xvecoo)(logx^n-[x])/([x]),w h e r en in N ,([x] denotes greatest integer less than or equal to x),t h e n-2L= , s. -1

If \ ^n C_r+\ ^n C_(r+1)=\ \ ^(n+1)C_x ,\ t h e n x = a. r b. r-1 c. n d. r+1

If S_n=sum_(r=0)^n 1/(nC_r) and t_n=sum_(r=0)^n r/(nC_r), then t_n/S_n=