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

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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Multiple Choice

Which of the following is an antiderivative of $f (x) = tan (e^{x} + 2)$ on $0 \leq x \leq ln (2)$ ?

\int tan (e^{x} + 2) d x = {sec}^{2} (e^{x} + 2) + C

\int tan (e^{x} + 2) d x = sin (e^{x} + 2) + C

\int tan (e^{x} + 2) d x = - ln | cos (e^{x} + 2) | + C

\int tan (e^{x} + 2) d x = tan (e^{x} + 2) + C

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Verified step by step guidance

Step 1: Recall the definition of an antiderivative. An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). In this case, we are tasked with finding the antiderivative of f(x) = tan(e^x + 2).

Step 2: Recognize that the derivative of -ln|cos(u)| is tan(u), where u is a function of x. This suggests that the antiderivative of tan(u) is -ln|cos(u) + C, where C is the constant of integration.

Step 3: Substitute u = e^x + 2 into the formula for the antiderivative. Since u is a composite function, we need to account for the chain rule when differentiating. The derivative of u with respect to x is du/dx = e^x.

Step 4: Verify that the derivative of -ln|cos(e^x + 2) with respect to x indeed equals tan(e^x + 2). Using the chain rule, differentiate -ln|cos(e^x + 2) to confirm that it simplifies to tan(e^x + 2).

Step 5: Conclude that the correct antiderivative of f(x) = tan(e^x + 2) is -ln|cos(e^x + 2) + C. This matches the given correct answer in the problem statement.