Applications

Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).

Find:

∫[f(x) + g(x)] dx

검증된 단계별 안내

First, identify the functions f(x) and g(x) as derivatives of given expressions. Specifically, f(x) = \(\frac{d}{dx}\) (1 - \(\sqrt{x}\)) and g(x) = \(\frac{d}{dx}\) (x + 2).

Next, compute f(x) by differentiating the function inside: recall that \(\sqrt{x}\) = x^{1/2}, so use the power rule for derivatives to find \(\frac{d}{dx}\) (1 - x^{1/2}).

Similarly, compute g(x) by differentiating the function inside: \(\frac{d}{dx}\) (x + 2), which involves differentiating a linear function.

After finding explicit expressions for f(x) and g(x), write the integral as \(\int\) [f(x) + g(x)] \, dx = \(\int\) f(x) \, dx + \(\int\) g(x) \, dx.

Finally, integrate each term separately. Since f(x) and g(x) are derivatives of known functions, integrating them will return the original functions (up to a constant). Combine the results and include the constant of integration.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative and Differentiation

The derivative of a function represents its instantaneous rate of change with respect to the variable. Differentiation rules, such as the power rule, allow us to find derivatives of functions like √x or polynomials. Understanding how to compute derivatives is essential to identify f(x) and g(x) in the problem.

Integration as the Inverse of Differentiation

Integration is the reverse process of differentiation, used to find the original function given its derivative. The integral of a sum of functions equals the sum of their integrals. Recognizing that ∫[f(x) + g(x)] dx can be simplified by integrating each term separately is key to solving the problem.

Properties of Definite and Indefinite Integrals

Indefinite integrals represent families of functions differing by a constant. When integrating derivatives, the result returns the original function plus a constant of integration. This concept helps in understanding that integrating f(x) + g(x), where f and g are derivatives, recovers the sum of the original functions.

ApplicationsSuppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).Find:∫[f(x) + g(x)] dx