Show : ` 2/ ln3 lt int_0^1 3^(x^(1/3)) dx lt int_0^1 4^(x^(1/3)) dx`

int_0^1 3^x dx

int_0^1 x^2(1-x)^(3/2) dx=

int_0^1(1)/(2x-3) dx .

int_0^1(dx)/(3x+2)

int_0^1 x(1 - x)^(3//2) dx =

(i) int_0^4 (2x+3) dx (ii) int_0^4 (2x-1)dx

int_0^1 |4x-3|dx

If I_(1) = int_0^(1) 2^(x^(2)) dx, 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,

show that (a) int_(0) ^(2pi) sin ^(3) x dx = 0 , (b) int_(-1)^(1) e^(-x^(2)) dx = 2 int_(0)^(1) e^(-x^(2)) dx