Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(2x³ − 5x + 7) dx

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Identify the integral to solve: $\int (2x^{3} - 5x + 7) \, dx$.

Recall the power rule for integration: $\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.

Apply the power rule to each term separately: integrate \$2x^{3}$, $-5x$, and $7$ individually.

For \$2x^{3}$, multiply the coefficient by the integral of $x^{3}$: \(2 \times \frac{x^{4}}{4}$; for \)-5x$, integrate $x$ as \(\frac{x^{2}}{2}$ and multiply by \)-5$; for the constant \(7$, integrate as \)7x$.

Combine all integrated terms and add the constant of integration $C$ to write the most general antiderivative.

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주요 개념

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Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed as a family of functions plus a constant of integration. It reverses differentiation and is written without limits, indicating all possible antiderivatives.

Power Rule for Integration

The power rule states that the integral of x^n (where n ≠ -1) is (x^(n+1))/(n+1) plus a constant. This rule is essential for integrating polynomial terms like x³ or x, allowing straightforward calculation of their antiderivatives.

Verification by Differentiation

After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors in the integration process.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫(2x³ − 5x + 7) dx