If f(x) and g(x) are integrable functions, th | vedclass

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(C) Let $F(x) = \int f(x) dx$ and $G(x) = \int g(x) dx$. Then $F'(x) = f(x)$ and $G'(x) = g(x)$.We want to find the expression for $F(x)G(x)$.By the p

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$\int [f(x) \int g(x) dx + g(x) \int f(x) dx] dx$

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(C) Let $F(x) = \int f(x) dx$ and $G(x) = \int g(x) dx$. Then $F'(x) = f(x)$ and $G'(x) = g(x)$.We want to find the expression for $F(x)G(x)$.By the product rule for differentiation, $\frac{d}{dx}[F(x)G(x)] = F(x)G'(x) + F'(x)G(x) = F(x)g(x) + f(x)G(x)$.Integrating both sides with respect to $x$, we get $F(x)G(x) = \int [f(x)G(x) + g(x)F(x)] dx$.Substituting back $F(x) = \int f(x) dx$ and $G(x) = \int g(x) dx$, we get $F(x)G(x) = \int [f(x) \int g(x) dx + g(x) \int f(x) dx] dx$.

If $f(x)$ and $g(x)$ are integrable functions, then which of the following is equal to $[\int f(x) dx][\int g(x) dx]$?

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