Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

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

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 $x^{3}$, \$5x$, and $-7$ individually.

For $x^{3}$, the integral is $\frac{x^{4}}{4}$; for \$5x$, treat the constant 5 as a multiplier and integrate $x$ to get \(\frac{x^{2}}{2}$, so the term becomes $5 \times \frac{x^{2}}{2}$; for \)-7$, integrate the constant to get $-7x$.

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

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed as a family of functions plus a constant of integration (C). 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) + C. This rule is essential for integrating polynomial terms like x³ and 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.

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Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

Textbook Question

Finding Indefinite Integrals