int_(0)^2(x^2+1)^2 d x

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This is a basic calculus question that requires the calculation of a definite integral. More precisely, we need to obtain the value of the integral of the function (x^2 + 1)^2 between the values x=0 and x=2. To achieve this, we first determine the anti-derivative of the given function, (x^2 + 1)^2. Secondly, we use the first part of the Fundamental Theorem of Calculus, which provides a method for evaluating definite integrals in terms of an antiderivative of its integrand. Specifically, it indicates that if F(x) is an antiderivative of f(x) on the interval [a, b], then the definite integral from a to b of f(x) is F(b) - F(a). We just find the antiderivative, replace x by 2 and then x by 0, before subtracting the two obtained values.