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.

∫(3t² + t/2) dt

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Identify the integral to solve: \(\int \left(3t^{2} + \frac{t}{2}\right) \, dt\).

Split the integral into the sum of two separate integrals: \(\int 3t^{2} \, dt + \int \frac{t}{2} \, dt\).

Apply the power rule for integration to each term separately. Recall that \(\int t^{n} \, dt = \frac{t^{n+1}}{n+1} + C\) for \(n \neq -1\).

Integrate the first term: \(\int 3t^{2} \, dt = 3 \int t^{2} \, dt = 3 \cdot \frac{t^{3}}{3} = t^{3}\).

Integrate the second term: \(\int \frac{t}{2} \, dt = \frac{1}{2} \int t \, dt = \frac{1}{2} \cdot \frac{t^{2}}{2} = \frac{t^{2}}{4}\), then combine the results and add the constant of integration \(C\).

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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. It does not have specified limits, unlike definite integrals.

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 fundamental for integrating polynomial terms like t² or t/2 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.∫(3t² + t/2) dt