Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Previous problemNext problem

3:20 minutes

Problem 4.9.67

Textbook Question

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

∫ (√x(2x⁶ - 4³√)dx

Verified step by step guidance

Rewrite the integral to simplify the expression. Break down the terms inside the integral: \( \int \sqrt{x}(2x^6 - 4\sqrt[3]{x}) \, dx \). Recall that \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{x} = x^{1/3} \). Distribute \( x^{1/2} \) across the terms inside the parentheses.

After distributing, the integral becomes \( \int (2x^{6 + 1/2} - 4x^{1/2 + 1/3}) \, dx \). Simplify the exponents by adding the powers: \( 6 + 1/2 = 13/2 \) and \( 1/2 + 1/3 = 3/6 + 2/6 = 5/6 \). The integral is now \( \int (2x^{13/2} - 4x^{5/6}) \, dx \).

Apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \). Integrate each term separately. For the first term, \( 2x^{13/2} \), the new exponent is \( 13/2 + 1 = 15/2 \), and the coefficient becomes \( \frac{2}{15/2} = \frac{4}{15} \). For the second term, \( -4x^{5/6} \), the new exponent is \( 5/6 + 1 = 11/6 \), and the coefficient becomes \( \frac{-4}{11/6} = \frac{-24}{11} \).

Combine the results of the integration. The indefinite integral becomes \( \frac{4}{15}x^{15/2} - \frac{24}{11}x^{11/6} + C \), where \( C \) is the constant of integration.

Check your work by differentiating the result. Differentiate \( \frac{4}{15}x^{15/2} - \frac{24}{11}x^{11/6} + C \) term by term using the power rule for differentiation: \( \frac{d}{dx}x^n = nx^{n-1} \). Verify that the derivative matches the original integrand \( \sqrt{x}(2x^6 - 4\sqrt[3]{x}) \).

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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 with the integral sign and do not have specified limits, resulting in a general solution plus a constant of integration (C). Understanding how to compute indefinite integrals is crucial for solving problems in calculus, as they are foundational to the concept of antiderivatives.

Integration Techniques

Various techniques are used to evaluate integrals, including substitution, integration by parts, and recognizing standard forms. For the given integral, recognizing the structure of the integrand, such as factoring or simplifying expressions, can facilitate the integration process. Mastery of these techniques is essential for effectively solving more complex integrals.

Verification by Differentiation

After finding an indefinite integral, it is important to verify the result by differentiating the antiderivative obtained. This process ensures that the derivative of the antiderivative returns to the original integrand. This verification step is a critical part of the integration process, confirming the accuracy of the solution and reinforcing the relationship between differentiation and integration.