23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (5s + 3)² ds

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Step 1: Recognize that the integral ∫ (5s + 3)² ds involves a polynomial raised to a power. To simplify, expand the square (5s + 3)² using the distributive property: (5s + 3)² = (5s)² + 2(5s)(3) + 3².

Step 2: Expand the terms: (5s)² = 25s², 2(5s)(3) = 30s, and 3² = 9. Combine these to rewrite the integrand as 25s² + 30s + 9.

Step 3: Break the integral into separate terms: ∫ (25s² + 30s + 9) ds = ∫ 25s² ds + ∫ 30s ds + ∫ 9 ds.

Step 4: Apply the power rule for integration to each term. For ∫ sⁿ ds, the rule is ∫ sⁿ ds = (sⁿ⁺¹)/(n+1) + C, where n ≠ -1. For constants, ∫ k ds = k * s + C.

Step 5: Combine the results of the integration for each term, ensuring to include the constant of integration (C). Finally, check your work by differentiating the result to verify it matches the original integrand.

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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function whose derivative matches the given function.

Integration Techniques

To solve integrals like ∫ (5s + 3)² ds, one may need to apply specific integration techniques. In this case, expanding the integrand before integrating can simplify the process. Techniques such as substitution or integration by parts may also be useful for more complex integrals, but recognizing when to apply these methods is key.

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiation. This involves taking the derivative of the antiderivative obtained and checking if it equals the original integrand. This step ensures that the integration was performed correctly and reinforces the fundamental relationship between differentiation and integration.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.∫ (5s + 3)² ds