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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2:05 minutes

Problem 8.1.8

Textbook Question

7–64. Integration review Evaluate the following integrals.
8. ∫ (9x - 2)^(-3) dx

Verified step by step guidance

Step 1: Recognize that the integral involves a power of a linear function, (9x - 2)^(-3). To solve this, use the substitution method. Let u = 9x - 2, which simplifies the expression.

Step 2: Compute the derivative of u with respect to x. Since u = 9x - 2, we find that du/dx = 9, or equivalently, dx = du/9.

Step 3: Rewrite the integral in terms of u. Substituting u and dx into the integral, it becomes ∫ u^(-3) * (1/9) du.

Step 4: Factor out the constant 1/9 from the integral. The integral now simplifies to (1/9) ∫ u^(-3) du.

Step 5: Apply the power rule for integration. Recall that ∫ u^n du = (u^(n+1))/(n+1) for n ≠ -1. Here, n = -3, so the integral becomes (1/9) * (u^(-3+1))/(-3+1). Simplify the expression and substitute back u = 9x - 2 to express the result in terms of x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating the area under a curve represented by a function over a specified interval. Understanding integration is crucial for evaluating integrals, as it allows us to reverse the process of differentiation.

Power Rule for Integration

The Power Rule for Integration is a technique used to integrate functions of the form x^n, where n is a real number. According to this rule, the integral of x^n is (x^(n+1))/(n+1) + C, provided n is not equal to -1. This rule is particularly useful when dealing with polynomial expressions and can be adapted for functions involving negative exponents.

Substitution Method

The Substitution Method is a technique used in integration to simplify the process by changing variables. It involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. This method is especially effective when dealing with composite functions or when the integrand contains a function and its derivative.