Prove that `pi/6 lt int_0^1(dx) /(sqrt(4-x^2-x^3)) lt pi/(4sqrt(2))`

Prove that 0 lt int_0^1(x^7dx)/((1+x^8)^(1/3)) lt 1/8

P rov et h a t1/2lt=int_0^(1/2)(dx)/(sqrt(1-x^(2n)))lt=pi/6forngeq1.

Prove that tan (sin^(-1)x) = x/(sqrt(1-x^(2))) - 1 lt x lt 1 .

Prove that pi/2 le sin ^(-1) x +2 cos^(-1) x lt (3 pi)/(2)

Prove that int_(0)^(pi/4)(sin2xdx)/(sin^(4)x+cos^4 x)=(pi)/(4)

If (1)/(sqrt2) lt x lt 1 , then prove that cos^(-1) x + cos^(-1) ((x + sqrt(1 - x^(2)))/(sqrt2)) = (pi)/(4)

Prove that sin^(-1). ((x + sqrt(1 - x^(2))/(sqrt2)) = sin^(-1) x + (pi)/(4) , where - (1)/(sqrt2) lt x lt(1)/(sqrt2)

If x lt 0 , the prove that cos^(-1) ((1 + x)/(sqrt(2(1 + x^(2))))) = (pi)/(4) - tan^(-1) x