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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2:13 minutes

Problem 8.1.10

Textbook Question

7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx

Verified step by step guidance

Step 1: Recognize that the integral involves an exponential function, e^(3 - 4x). To simplify, identify the inner function (3 - 4x) and its derivative (-4). This suggests using substitution.

Step 2: Perform substitution. Let u = 3 - 4x, which implies that du/dx = -4 or equivalently, dx = -du/4.

Step 3: Rewrite the integral in terms of u. Substituting u and dx, the integral becomes ∫ e^u * (-du/4). Factor out the constant -1/4 to simplify: (-1/4) ∫ e^u du.

Step 4: Integrate e^u with respect to u. The integral of e^u is simply e^u, so the result becomes (-1/4) * e^u + C, where C is the constant of integration.

Step 5: Substitute back u = 3 - 4x to return to the original variable. The final expression is (-1/4) * e^(3 - 4x) + C.

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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 determining 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.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus, especially in integration problems involving e raised to a power. Recognizing the structure of exponential functions is essential for applying integration techniques.

Substitution Method

The substitution method is a technique used in integration to simplify the process of finding an integral. It involves substituting a part of the integrand with a new variable to make the integral easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative, allowing for a more straightforward integration process.