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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1:49 minutes

Problem 8.1.53

Textbook Question

7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx

Verified step by step guidance

Step 1: Recognize that the integral involves a composite function, eˣ, inside another function sec(eˣ + 1). This suggests that substitution might be a useful technique.

Step 2: Let u = eˣ + 1. Then, compute the derivative of u with respect to x: du/dx = eˣ. This implies that du = eˣ dx.

Step 3: Rewrite the integral in terms of u. Substituting u = eˣ + 1 and du = eˣ dx, the integral becomes ∫ sec(u) du.

Step 4: Recall the standard integral formula for sec(u): ∫ sec(u) du = ln|sec(u) + tan(u)| + C, where C is the constant of integration.

Step 5: Substitute back u = eˣ + 1 into the result to express the solution in terms of x. The final expression will be ln|sec(eˣ + 1) + tan(eˣ + 1)| + 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. Common techniques include substitution, integration by parts, and partial fractions. In this case, recognizing the structure of the integrand can help determine the appropriate method to simplify the integral.

Substitution Method

The substitution method is a powerful technique in integration where a new variable is introduced to simplify the integral. By letting u = eˣ + 1, the integral can be transformed into a more manageable form. This method is particularly useful when the integrand contains a function and its derivative.

Exponential Functions

Exponential functions, such as eˣ, are functions of the form f(x) = a^x, where 'a' is a constant. They have unique properties, including the fact that their derivative is equal to the function itself. Understanding the behavior of exponential functions is crucial for evaluating integrals involving them, as they often appear in various calculus problems.