The integral expression of the form ∫ f(x) · dg is equal to the composite expression f(x) · g(x) - ∫ g(x) df by applying integration by parts method. (Correct choice: A)
How to derive the rule of integration by parts
In this question we must derive a integration rule for the product of two functions that are integrable. The integration by parts method is inspired on the differentiation rule for the product of two differentiable functions, whose definition is shown below:
[tex]\frac{d}{dx} [f(x)\cdot g(x)] = f'(x) \cdot g(x) + f(x)\cdot g'(x)[/tex] (1)
If we integrate and used integral Calculus theorem on both sides of (1), then we have the following expression:
f(x) · g(x) = ∫ g(x) · df + ∫ f(x) · dg
∫ f(x) · dg = f(x) · g(x) - ∫ g(x) df
Therefore, the integral expression of the form ∫ f(x) · dg is equal to the composite expression f(x) · g(x) - ∫ g(x) df by applying integration by parts method. (Correct choice: A)
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