Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x² − 2x + 1

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Identify the function to find the antiderivative of: \(x^{2} - 2x + 1\).

Recall that the antiderivative (indefinite integral) of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\). We will integrate each term separately.

Use the power rule for integration: For any term \(x^{n}\), the antiderivative is \(\frac{x^{n+1}}{n+1}\), provided \(n \neq -1\).

Integrate each term: \(\int x^{2} \, dx = \frac{x^{3}}{3}\), \(\int (-2x) \, dx = -2 \cdot \frac{x^{2}}{2} = -x^{2}\), and \(\int 1 \, dx = x\).

Combine the results and add the constant of integration \(C\): \(F(x) = \frac{x^{3}}{3} - x^{2} + x + C\). This is the general antiderivative of the given function.

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주요 개념

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Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is often expressed with an arbitrary constant, C, since differentiation of a constant is zero.

Power Rule for Integration

The power rule for integration states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) + C. This rule is essential for integrating polynomial terms like x² and x.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps avoid mistakes in integration.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.x² − 2x + 1