Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁷

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Recall that an antiderivative of a function is another function whose derivative is the original function. For power functions, the antiderivative rule is: if \(f(x) = x^n\), then an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration and \(n \neq -1\).

Identify the exponent in the given function \(x^7\). Here, \(n = 7\).

Apply the antiderivative formula by increasing the exponent by 1: \(7 + 1 = 8\).

Divide by the new exponent to get the antiderivative: \(\frac{x^8}{8} + C\).

Verify your answer by differentiating \(\frac{x^8}{8} + C\) and checking that the derivative is \(x^7\).

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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 fundamental for finding antiderivatives of polynomial functions like x⁷.

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.x⁷