Expert Guide: How to Integrate the Expression ∫ [f(u) ± g(u)] du

∫ [f(u) ± g(u)] du

To integrate the expression ∫ [f(u) ± g(u)] du, we can first evaluate the integrals of f(u) and g(u) separately, and then combine the results

To integrate the expression ∫ [f(u) ± g(u)] du, we can first evaluate the integrals of f(u) and g(u) separately, and then combine the results.

Let’s start by evaluating the integral of f(u). The integral of f(u) with respect to u can be denoted as ∫ f(u) du. This is the antiderivative of f(u) and can be found using various integration techniques, such as basic rules of integration, substitution, or integration by parts. Once we find the antiderivative, we can represent it as F(u) + C, where F(u) is the antiderivative function and C is the constant of integration.

Next, we evaluate the integral of g(u), denoted as ∫ g(u) du. Similar to f(u), we apply integration techniques to find the antiderivative of g(u). The result will be G(u) + C, where G(u) is the antiderivative function of g(u) and C is the constant of integration.

Now, let’s combine the integrals of f(u) and g(u), considering the ± sign.

If the ± sign is a plus (+), the expression becomes ∫ [f(u) + g(u)] du. In this case, we can add the antiderivatives F(u) + G(u) and include the constant of integration to obtain the final result: F(u) + G(u) + C. This represents the integral of [f(u) + g(u)] with respect to u.

If the ± sign is a minus (-), the expression becomes ∫ [f(u) – g(u)] du. In this situation, we subtract the antiderivatives F(u) – G(u) and add the constant of integration to get: F(u) – G(u) + C. This represents the integral of [f(u) – g(u)] with respect to u.

Remember, the constant of integration (C) is added because taking the derivative of a constant yields zero, and we need to account for all possible antiderivatives.

In summary,

∫ [f(u) ± g(u)] du =
– If ± is a plus (+): F(u) + G(u) + C
– If ± is a minus (-): F(u) – G(u) + C

Note that the specific form of F(u) and G(u) will depend on the functions f(u) and g(u), and you may need to use integration techniques to determine their antiderivatives.

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