If `I_K=int_1^e(lnx)^kdx(k in I^+)dx(k in I^+),` then find the value of `I_4dot`

If int_0^1(e^(-x)dx)/(1+e^x)=(log)_e(1+e)+k , then find the value of k.

If |z|lt=4 , then find the maximum value of |i z+3-4i|dot

If I_n=int( lnx)^n dx then I_n+nI_(n-1)

Find the real part of (1-i)^(-i)dot

The position vectors of the point A ,B ,Ca n dDa r e3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k ,- hat i+ hat j+2 hat k and 4 hat i+5 hat j+lambda hat k , respectively. If the points A ,B ,Ca n dD lie on a plane, find the value of lambdadot

IfI_I=int_0^(pi//2)cos(sinx)dx ,I_2=int_0^(pi/2)sin(cosx)d ,a n dI_3=int_0^(pi/2)cosx dx , then find the order in which the values I_1,I_2,I_3, exist.

IfI_1=int_0^1 2^x^3 dx,I_2=int_0^1 2^x^2 dx ,I_3=int_1^2 2^x^2dx ,I_4=int_1^2 2^x^3dx , then which of the following is/are ture? I_1> I_2 (b) I_2> I_2 I_3> I_4 (d) I_3

IfI_1=int_0^1 2^(x^2) ,I_2=int_0^1 2^(x^3)dx ,I_3=int_1^2 2^(x^2)dx ,I_4=int_1^2 2^(x^3)dx , then which of the following is/are true? I_1> I_2 (b) I_2> I_1 I_3> I_4 (d) I_3 lt I_4

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

If vectors vec a= hat i+2 hat j- hat k , vec b=2 hat i- hat j+ hat ka n d vec c="lambda" hat"i"+ hat"j"+2 hat"k" are coplanar, then find the value of (lambda-4)dot