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


∫ (x² - 36) / (x - 6) dx

Verified step by step guidance

1. Rewrite the integrand by performing polynomial long division on \( \frac{x^2 - 36}{x - 6} \). Start by dividing the leading term \( x^2 \) by \( x \), which gives \( x \). Multiply \( x \) by \( x - 6 \) and subtract from \( x^2 - 36 \). This simplifies the integrand.
2. After performing the division, the integrand simplifies to \( x + 6 \). Rewrite the integral as \( \int (x + 6) \, dx \).
3. Split the integral into two separate terms: \( \int x \, dx + \int 6 \, dx \).
4. Apply the power rule for integration to each term. For \( \int x \, dx \), use the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) with \( n = 1 \). For \( \int 6 \, dx \), treat 6 as a constant and integrate it as \( 6x + C \).
5. Combine the results of the two integrals to write the final expression for the indefinite integral. Don't forget to include the constant of integration \( C \). Finally, verify your result by differentiating it to ensure 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 whose derivative matches the given function.

Polynomial Long Division

When integrating rational functions, polynomial long division is used to simplify the integrand if the degree of the numerator is greater than or equal to the degree of the denominator. This process involves dividing the numerator by the denominator to express the integrand as a sum of a polynomial and a proper fraction, making it easier to integrate.

Checking Work by Differentiation

To verify the correctness of an indefinite integral, one can differentiate the result. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be correct. This step is crucial in calculus to ensure that the integration process has been performed accurately.