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


∫ (3x + 1) (4 - x) dx

Verified step by step guidance

1. Step 1: Expand the integrand (3x + 1)(4 - x) using the distributive property. Multiply each term in the first parenthesis by each term in the second parenthesis to rewrite the integrand.
2. Step 2: Combine like terms after expanding. This will simplify the integrand into a polynomial expression.
3. Step 3: Apply the power rule for integration to each term of the polynomial. Recall that the integral of x^n is (x^(n+1))/(n+1) and the integral of a constant is the constant multiplied by x.
4. Step 4: Add the constant of integration, C, to the result since this is an indefinite integral.
5. Step 5: Check your work by differentiating the result. Use the derivative rules to ensure that the derivative of your solution 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 by Expansion

Integration by expansion involves simplifying the integrand before integrating. This can be done by distributing or expanding the terms within the integral. For example, in the integral ∫ (3x + 1)(4 - x) dx, we first expand the product to obtain a polynomial, which can then be integrated term by term.

Checking Work 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 ensuring it matches the original integrand. This step confirms the correctness of the integration process and helps identify any potential errors in calculations.