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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Multiple Choice

Evaluate the indefinite integral. (Use c for the constant of integration.)

$\int_{}^{} \frac{s i n (t)}{1 + c o s (t)} d t$

t a n (\frac{t}{2}) + c

- l n | 1 + c o s (t) | + c

l n | s i n (t) | + c

s e c (t) + c

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Verified step by step guidance

Step 1: Recognize that the integral involves a trigonometric function and consider using a substitution to simplify the expression. A common substitution for integrals involving sin(t) and cos(t) is the Weierstrass substitution: let u = tan(t/2), which transforms trigonometric functions into rational functions.

Step 2: Recall the relationships for the Weierstrass substitution: sin(t) = 2u / (1 + u^2) and cos(t) = (1 - u^2) / (1 + u^2). Also, dt = 2 du / (1 + u^2). Substitute these into the integral.

Step 3: Replace sin(t), cos(t), and dt in the integral with their expressions in terms of u. The integral becomes ∫ (2u / (1 + u^2)) / (1 + (1 - u^2) / (1 + u^2)) * (2 du / (1 + u^2)). Simplify the expression to make the integral easier to evaluate.

Step 4: Simplify the denominator and numerator to reduce the integral to a simpler form. After simplification, the integral should be expressed in terms of u, which can be integrated using standard techniques.

Step 5: Perform the integration in terms of u, then substitute back u = tan(t/2) to return the solution in terms of t. Add the constant of integration, c, to complete the indefinite integral.