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


∫ (3/x⁴ + 2 - 3/x²)

Verified step by step guidance

1. Step 1: Break down the integral into separate terms. The given integral is ∫(3/x⁴ + 2 - 3/x²) dx. Rewrite it as ∫(3/x⁴) dx + ∫2 dx - ∫(3/x²) dx.
2. Step 2: Apply the power rule for integration to each term. Recall that the power rule states ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. For terms involving x raised to negative powers, adjust the exponent accordingly.
3. Step 3: For the first term, ∫(3/x⁴) dx, rewrite it as ∫3x⁻⁴ dx. Using the power rule, integrate to get (3 * x⁻³)/-3 = -x⁻³.
4. Step 4: For the second term, ∫2 dx, integrate the constant to get 2x.
5. Step 5: For the third term, ∫(3/x²) dx, rewrite it as ∫3x⁻² dx. Using the power rule, integrate to get (3 * x⁻¹)/-1 = -3x⁻¹. Combine all results and add the constant of integration, C.

Key Concepts

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, various techniques can be employed, such as substitution, integration by parts, and recognizing standard forms. In this case, the integrand consists of rational functions, which can often be integrated term by term. Understanding how to manipulate and simplify expressions is crucial for effectively applying these techniques.

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative. This process ensures that the derivative of the obtained function returns to the original integrand. This step is a critical part of the integration process, confirming the correctness of the integration performed.