Prove that `int_0^1sqrt((1+x)(1+x^3))dx`cannot exceed `sqrt((15)/8)`.

1.	Prove that `int_0^1sqrt((1+x)(1+x^3))dx`cannot exceed `sqrt((15)/8)`.
Answer» `int_(0)^(1)sqrt((1+x)(1+x^(3)))dx le sqrt((int_(0)^(1)(1+x)dx)(int_(0)^(1)(1+x^(3))dx))` `=sqrt((x+(x^(2))/2)_(0)^(1)(x+(x^(4))/4)_(0)^(1))` `=sqrt((3/2)(5/4))` `=sqrt(15/8)`

Discussion

Consider the integral `I=int_(0)^(2pi)(dx)/(5-2cosx)` Making the substitution `"tan"1/2x=t`, we have `I=int_(0)^(2pi)(dx)/(5-2cosx)=int_(0)^(0)(2dt)/((1+t^(2))[5-2(1-t^(2))//(1+t^(2))])=0` The result is obviously wrong, since the integrand is positive and consequently the integral of this function cannot be equal to zero. Find the mistake.
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`lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))`
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