`F i n d t h e v a l u e o fxif1/(a+b+x)=1/a+1/b+1/2`

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))

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

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

If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b ,t h e n a is e q u a l 1/3 (b) 2/3 (c) -1/3 (d0 -2/3

If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b ,t h e n a is e q u a l 1/3 (b) 2/3 (c) -1/3 (d0 -2/3

Let L=("lim")_(xvec0)(a-sqrt(a^2-x^2)-(x^2)/4)/(x^4),a > 0. IfLi sfin i t e ,t h e n a=2 (b) a=1 L=1/(64) (d) L=1/(32)

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

If sin(A-B)=1/(sqrt(10 ,))cos(A+B)=2/(sqrt(29)),fin dt h ev a l u eof2Aw h e r eAa n dBl i eb e t w e e n0a n dpi/4dot

If f(x)=(e^x)/(1+e^x), I_1=int_(f(-a))^(f(a))xg(x(1-x)dx and I_2=int_(f(-a))^(f(a)) g(x(1-x))dx , then the value of (I_2)/(I_1) is (a) -1 (b) -2 (c) 2 (d) 1

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