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.

∫(1/x² − x² − 1/3) dx

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Rewrite the integral by expressing each term with exponents: \(\int \left( x^{-2} - x^{2} - \frac{1}{3} \right) \, dx\).

Use the linearity of the integral to split it into separate integrals: \(\int x^{-2} \, dx - \int x^{2} \, dx - \int \frac{1}{3} \, dx\).

Apply the power rule for integration to each term: For \(\int x^{n} \, dx\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).

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

Combine the results, remembering to include a single constant of integration \(+ C\) at the end, and write the most general antiderivative.

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주요 개념

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Indefinite Integral

An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation and is written as ∫f(x) dx, yielding a function F(x) such that F'(x) = f(x).

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating polynomial terms by increasing the exponent by one and dividing by the new exponent.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral solution 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.∫(1/x² − x² − 1/3) dx