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


∫ ((4x⁴ - 6x²) / x ) dx

Verified step by step guidance

1. Rewrite the integrand by simplifying the expression. Divide each term in the numerator by the denominator x: \( \frac{4x^4}{x} - \frac{6x^2}{x} = 4x^3 - 6x \). The integral becomes \( \int (4x^3 - 6x) \, dx \).
2. Apply the power rule for integration to each term. The power rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \).
3. Integrate the first term \( 4x^3 \): Using the power rule, \( \int 4x^3 \, dx = \frac{4x^{3+1}}{3+1} = x^4 \).
4. Integrate the second term \( -6x \): Using the power rule, \( \int -6x \, dx = \frac{-6x^{1+1}}{1+1} = -3x^2 \).
5. Combine the results of the integration: The indefinite integral is \( x^4 - 3x^2 + C \), where \( C \) is the constant of integration. To check your work, differentiate \( x^4 - 3x^2 + C \) and verify that it equals 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.

Simplifying Expressions

Before integrating, it is often necessary to simplify the integrand. This can involve algebraic manipulation, such as dividing terms or factoring. In the given integral, simplifying the expression (4x⁴ - 6x²) / x leads to a more straightforward form, allowing for easier integration of each term separately.

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 the calculations.