Prove that `ln 2 lt int_(0)^(1)(dx)/(sqrt(1+x^(4))) lt (pi)/2)`

Prove that : int_(0)^(1) (log x)/(sqrt(1-x^(2)))dx=-(pi)/(2)log 2

Prove that 1 lt int_(0)^(2)((5-x)/(9-x^(2)))dx lt 6/5

For x in(0,1) arrange f_(1)(x) = (1)/(9-x^(2)), f_(2)(x) = (1)/(9-2x^(2)) and f_(3)(x) = (1)/(9-x^(2)-x^(3)) in ascending order and hence prove that 1/6 ln2 lt int_(0)^(1)(1)/(9-x^(2)-x^(3)) dx lt (1)/(6sqrt(2)) ln 5 .

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

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

Show that (i) (1)/(10sqrt(2))lt underset(0)overset(1)int(x^(9))/(sqrt(1+x))dx lt 1/10 , (ii) 1/2 ln 2 lt int_(0)^(1)(tanx)/(1+x^(2))dx lt (pi)/(2)

Prove that tan^(-1) {(x)/(a + sqrt(a^(2) - x^(2)))} = (1)/(2) sin^(-1).(x)/(a), -a lt x lt a

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

Prove that :int_(0)^(pi) (x)/(1 +sin^(2) x) dx =(pi^(2))/(2sqrt(2))