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.

∫(x + 1) dx

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Identify the integral to solve: \(\int (x + 1) \, dx\).

Recall the basic rule for integrating sums: \(\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx\).

Split the integral into two separate integrals: \(\int x \, dx + \int 1 \, dx\).

Use the power rule for integration on the first term: \(\int x \, dx = \frac{x^{2}}{2} + C_1\), where \(C_1\) is a constant of integration.

Integrate the constant term: \(\int 1 \, dx = x + C_2\), where \(C_2\) is another constant of integration. Combine constants into a single constant \(C\) to write the 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, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C.

Basic Integration Rules

Integration rules such as the power rule and linearity simplify finding antiderivatives. For example, ∫x dx = x²/2 + C and ∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx, allowing term-by-term integration.

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 ensures no 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.∫(x + 1) dx