Let int_0^1 f(x) , dx = 1,int_0^1 x f(x) , dx | vedclass

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(A) We are given the integrals: $\int_0^1 f(x) \, dx = 1$,$\int_0^1 x f(x) \, dx = a$,$\int_0^1 x^2 f(x) \, dx = a^2$. We need to evaluate the integra

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(A) We are given the integrals: $\int_0^1 f(x) \, dx = 1$,$\int_0^1 x f(x) \, dx = a$,$\int_0^1 x^2 f(x) \, dx = a^2$. We need to evaluate the integral $I = \int_0^1 (x - a)^2 f(x) \, dx$. Expanding the term $(x - a)^2 = x^2 - 2ax + a^2$,we get: $I = \int_0^1 (x^2 - 2ax + a^2) f(x) \, dx$ $I = \int_0^1 x^2 f(x) \, dx - 2a \int_0^1 x f(x) \, dx + a^2 \int_0^1 f(x) \, dx$ Substituting the given values: $I = a^2 - 2a(a) + a^2(1)$ $I = a^2 - 2a^2 + a^2$ $I = 0$.

Let $\int_0^1 f(x) \, dx = 1$,$\int_0^1 x f(x) \, dx = a$,and $\int_0^1 x^2 f(x) \, dx = a^2$. Then the value of $\int_0^1 (x - a)^2 f(x) \, dx$ is:

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