Prove that `4 le int_(1)^(3)sqrt(3+x^(2))le4sqrt(3)`

Prove that tan ^(-1) ((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2)))=pi/4+1/2 cos ^(-1) x^2

Evaluate int(1)/((x-3)sqrt(x+1))dx .

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

If alpha_(1), alpha_(2), alpha_(3), alpha_(4) are the roots of the equation x^(4)+(2-sqrt(3))x^(2)+2+sqrt(3)=0 , then the value of (1-alpha_(1))(1-alpha_(2))(1-alpha_(3))(1-alpha_(4)) is :

Evaluate int(x+3)/sqrt(5-4x-x^2)dx

Find the value of int_(0)^(1)3sqrt(2x^(3)-3x^(2)-x+1)dx .

Find int (x+3) sqrt(3-4x-x^2) dx

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

prove tan ^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x)))=pi/4-1/2 cos ^(-1) x, -1/2 le x le 1