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

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Identify the function for which you need to find the antiderivative. Here, the function is \$2x$.

Recall that finding an antiderivative means finding a function $F(x)$ such that $F'(x) = 2x$.

Use the power rule for integration: the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.

Apply the power rule to \$2x$, which can be written as $2x^1$. Integrate to get $2 \times \frac{x^{1+1}}{1+1} + C$.

Simplify the expression to find the antiderivative function $F(x)$, and remember to add the constant of integration $C$.

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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 expressed with an arbitrary constant 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) plus a constant. This rule is essential for finding antiderivatives of polynomial functions like 2x.

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.2x