Investigate if function f(subscript n)(x) = integral (0 to x) t^n sqroot(t^2+1)dt increase?

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You need to test if the first derivative of the function is positive, hence, you need to differentiate the given function with respect to `x` , such that:

`f_n'(x) = (int_0^x t^n*sqrt(t^2 + 1)dt)' => f_n'(x) = x^n*sqrt(x^2 + 1)`

Since `x^2 > 0, AA x in R => x^2 + 1 > 0 => sqrt(x^2 + 1)>0` .

You need to notice though that the problem does not provide information about the domain of definition of the given function, because if the domain would be `[0,oo)` , then the derivative is strictly positive, but if the domain would be the set `R` , then the derivative is negative over `(-oo,0).`

Hence, testing if the given function increases, yields that the statement is valid only if `x >=0` .