Prove that `4le int_(1)^(3) sqrt(3+x^(3)) dx le 2sqrt30`.

Prove that 4 lt= int_1^3 sqrt(3+x^2) dx lt= 4sqrt(3)

Prove the following inequalities : (i) (sqrt(3))/(8) lt int_(pi//4)^(pi//3)(sinx)/(x)dx lt (sqrt(2))/6 , (ii) 4 le int_(1)^(3)sqrt((3x^(2)+x^(2)))dx le 20 sqrt(30)

int(1)/(sqrt(2x^(2)+3x-2))dx

int((1+x)^3)/(3sqrt(x^2))dx

int(1)/(sqrt(x^(2)+3x+1))dx

int(x+1)/(sqrt(2x^(2)+x-3))dx

int(1)/(sqrt(2+x-3x^2))dx

Prove that: int_(-a)^(a) x^(3) sqrt(a^(2) -x^(2) ) dx=0

int(1)/(sqrt(3-(x^(2))/(4))dx)

int (x^(2) +1)/(sqrt(x^(2)+3))dx