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

Integrals of Trig Functions

Calculus

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

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1:40 minutes

Problem 8.3.7

Textbook Question

7. How would you evaluate ∫ tan¹⁰x sec²x dx?

Verified step by step guidance

Recognize that the integral involves a trigonometric function raised to a power, specifically tan¹⁰(x), and sec²(x). Recall that the derivative of tan(x) is sec²(x), which suggests a substitution method might be appropriate.

Let u = tan(x). Then, the derivative of u with respect to x is du/dx = sec²(x), or equivalently, du = sec²(x) dx. This substitution simplifies the integral.

Rewrite the integral in terms of u. Since tan(x) = u and sec²(x) dx = du, the integral becomes ∫ u¹⁰ du.

Apply the power rule for integration, which states that ∫ uⁿ du = (uⁿ⁺¹)/(n+1) + C, where n ≠ -1. Here, n = 10, so the integral becomes (u¹¹)/11 + C.

Finally, substitute back u = tan(x) to express the result in terms of x. The final expression is (tan¹¹(x))/11 + C.

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Key Concepts

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. In this case, recognizing that the integral involves the function tan(x) and its derivative sec²(x) is crucial. This allows for the use of substitution, where we can let u = tan(x), simplifying the integral significantly.

Substitution Method

The substitution method is a powerful technique in calculus that simplifies the process of integration. By substituting a part of the integral with a new variable, we can transform the integral into a more manageable form. For the integral ∫ tan¹⁰x sec²x dx, substituting u = tan(x) leads to a straightforward integration of u¹⁰ du.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is essential in calculus. An indefinite integral, like ∫ tan¹⁰x sec²x dx, represents a family of functions and includes a constant of integration. In contrast, a definite integral computes the area under the curve between two limits. Recognizing this distinction helps in correctly interpreting the results of integration.