Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx
5
views
7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
2:13 minutesProblem 4.9.37
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (x² - 36) / (x - 6) dx
Verified step by step guidance1
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.
After performing the division, the integrand simplifies to \( x + 6 \). Rewrite the integral as \( \int (x + 6) \, dx \).
Split the integral into two separate terms: \( \int x \, dx + \int 6 \, dx \).
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 \( 6x + C \).
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:04
Introduction to Indefinite Integrals
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.
Recommended video:
07:00Taylor Polynomials
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.
Recommended video:
06:22Introduction To Work
5:04mWatch next
Master Introduction to Indefinite Integrals with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice