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

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3:00 minutes

Problem 4.9.65

Textbook Question

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

∫ ((1 + √x)/x)dx

Verified step by step guidance

Rewrite the integrand to simplify the expression. Split the fraction into two terms: \( \frac{1}{x} + \frac{\sqrt{x}}{x} \). Simplify further to get \( \frac{1}{x} + x^{-1/2} \).

Express the integrand in terms of powers of \(x\): \( x^{-1} + x^{-1/2} \). This makes it easier to apply the power rule for integration.

Apply the power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \(n \neq -1\). For \(x^{-1}\), recall that \( \int x^{-1} dx = \ln|x| + C \).

Integrate each term separately: \( \int x^{-1} dx = \ln|x| \) and \( \int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2} \).

Combine the results to write the final integral: \( \ln|x| + 2x^{1/2} + C \), where \(C\) is the constant of integration. Check your work by differentiating this result to ensure it matches the original integrand.

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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 without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation, where we seek a function F(x) such that F'(x) equals the integrand.

Integration Techniques

To solve integrals, various techniques can be employed, such as substitution, integration by parts, or partial fraction decomposition. In this case, simplifying the integrand, which is a rational function, can make the integration process more manageable. Recognizing patterns and applying appropriate techniques is crucial for finding the correct antiderivative.

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative obtained. This step ensures that the differentiation of the antiderivative returns the original integrand. This verification process is a fundamental practice in calculus, confirming the correctness of the integration performed.