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


∫ (4√x - (4 /√x)) dx

Verified step by step guidance

1. Step 1: Break down the integral into separate terms. The given integral is ∫ (4√x - (4/√x)) dx. Rewrite it as ∫ 4√x dx - ∫ (4/√x) dx to handle each term individually.
2. Step 2: Express the square root terms in terms of exponents. Recall that √x = x^(1/2) and 1/√x = x^(-1/2). Substitute these into the integral: ∫ 4x^(1/2) dx - ∫ 4x^(-1/2) dx.
3. Step 3: Apply the power rule for integration to each term. The power rule states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. For the first term, integrate 4x^(1/2) using the power rule. For the second term, integrate 4x^(-1/2) using the same rule.
4. Step 4: Combine the results of the integrations. After applying the power rule, you will have expressions for both terms. Add them together to form the complete indefinite integral.
5. Step 5: Add the constant of integration, C, to the final result. This constant accounts for the family of antiderivatives. To verify your work, differentiate the result and check if it matches the original integrand.

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 F(x) such that F'(x) equals the integrand.

Integration Techniques

To solve integrals, various techniques can be employed, including substitution, integration by parts, and recognizing standard forms. In this case, the integrand consists of terms involving powers of x, which can be integrated using the power rule. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, applicable for n ≠ -1.

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 crucial for confirming the correctness of the integration process and helps identify any potential errors in the calculation.