`int_(z=0) ^ 1 int_(y=0) ^ z int_(x=0)^y xy^2z^3 dxdydz` is

int_(z=0)^(1)int_(y=0)^(z)int_(x=0)^(y)xy^(2)z^(3)dxdydz is

int_(0)^(2)int_(0)^(x)int_(0)^(y)xydxdydz

int_(-1)^(1)int_(0)^(2)int_(x-2)^(x+2)(x+y+z)dxdydz

(v) int_(x=0)^(1)int_(y=0)^(1)(dxdy)/((x+y+1)^(2))

Let y = f(x) be a differentiable function, AA x in R and satisfies, f(x) = x+ int_(0)^(1)x^(2)z f(z) dz + int_(0)^(1)x z^(2) f(z) dz , then

Find the derivative (dy)/(dx) of the following implicit functions : (a) int_(0)^(y) e^(-t^(2)) dt + int_(0)^(x^(3)) sin ^(2) t dt = 0 (b) int_(0)^(y) e^(t) dt + int _(0)^(x) sin t dt = 0 (c) int _(pi//2)^(x) sqrt(3 - 2 sin^(2)z ) dz + int_(0)^(y) cos t dt = 0

If int_(0)^(1)f(x)dx=1 and int_(1)^(2)f(y)dy=2 , then int_(0)^(2)f(z)dz=

If int_(0)^(x)f(z)dz=x+int_(x)^(1)zf(z)dz , then int_(1)^(2)f(x)dx equals