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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3:19 minutes

Problem 8.7.6

Textbook Question

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

Verified step by step guidance

Recognize that the integral involves a trigonometric function and a square root expression. The substitution u = sin(x) is suggested to simplify the integral. Start by calculating the derivative of u: du = cos(x) dx.

Substitute u = sin(x) into the integral. Replace sin²(x) with u² and cos(x) dx with du. The integral becomes ∫ √(100 − u²) du.

Notice that the integral ∫ √(a² − u²) du matches a standard form in integral tables. Specifically, it corresponds to the formula for ∫ √(a² − u²) du = (1/2) [u √(a² − u²) + a² arcsin(u/a)] + C, where a is a constant.

Identify the constant a in the given integral. Here, a² = 100, so a = 10. Substitute a = 10 into the standard formula.

After applying the formula, back-substitute u = sin(x) to express the result in terms of x. This completes the evaluation of the integral.

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Key Concepts

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

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of expressions containing trigonometric functions. By substituting a trigonometric function for a variable, we can transform the integral into a more manageable form. In this case, substituting u = sin x allows us to express the integral in terms of u, simplifying the evaluation process.

Integration by Substitution

Integration by substitution is a method that simplifies the process of finding integrals by changing the variable of integration. This technique involves substituting a new variable, which often makes the integral easier to evaluate. In the given question, the substitution u = sin x will help in transforming the integral into a form that can be evaluated using standard integral tables.

Integral Tables

Integral tables are collections of pre-calculated integrals that provide quick references for evaluating common integrals. They are particularly useful when dealing with complex functions that do not have straightforward antiderivatives. After performing the substitution in the given integral, one can refer to these tables to find the integral of the resulting expression, facilitating a faster solution.

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Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

37. ∫ dx / √(x² + 10x), x >

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