Applications

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

Find:

∫f(x) dx

검증된 단계별 안내

First, identify the functions f(x) and g(x) given in the problem. We have f(x) = \(\frac{d}{dx}\) (1 - \(\sqrt{x}\)) and g(x) = \(\frac{d}{dx}\) (x + 2).

Next, compute the derivative f(x) by differentiating the function inside the derivative: 1 - \(\sqrt{x}\). Recall that \(\sqrt{x}\) = x^{1/2}, so use the power rule for differentiation.

After finding f(x), set up the integral \(\int\) f(x) \, dx. Since f(x) is the derivative of (1 - \(\sqrt{x}\)), integrating f(x) with respect to x will give you the original function plus a constant of integration.

Write the integral result as (1 - \(\sqrt{x}\)) + C, where C is the constant of integration, because integration is the inverse operation of differentiation.

Finally, verify your result by differentiating your integral expression to ensure it matches f(x). This confirms the correctness of your 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 the rate at which the function's value changes with respect to its variable. Differentiation rules, such as the power rule, help find derivatives of expressions like √x or x + 2. Understanding how to compute derivatives is essential to identify f(x) and g(x) as given.

Antiderivative and Indefinite Integral

The antiderivative or indefinite integral of a function reverses differentiation, finding a function whose derivative is the given function. The integral symbol ∫f(x) dx represents all functions whose derivative is f(x), plus a constant of integration. This concept is key to solving ∫f(x) dx.

Power Rule for Integration

The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is used to integrate functions involving powers of x, such as those derived from derivatives of √x or polynomials. Applying this rule simplifies finding the antiderivative of f(x).

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