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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1:16 minutes

Problem 4.9.45

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (sec² Θ + sec Θ tan Θ)dΘ

Verified step by step guidance

Rewrite the integral as the sum of two separate integrals: ∫(sec²Θ)dΘ + ∫(secΘ tanΘ)dΘ. This is possible because the integral of a sum is the sum of the integrals.

Recognize the standard integral formulas: ∫(sec²Θ)dΘ = tanΘ + C₁ and ∫(secΘ tanΘ)dΘ = secΘ + C₂, where C₁ and C₂ are constants of integration.

Combine the results of the two integrals: tanΘ + secΘ + C, where C is the combined constant of integration (C = C₁ + C₂).

Verify your solution by differentiating the result, tanΘ + secΘ + C. Use the derivative rules: d/dΘ(tanΘ) = sec²Θ and d/dΘ(secΘ) = secΘ tanΘ.

Check that the derivative of tanΘ + secΘ + C matches the original integrand, sec²Θ + secΘ tanΘ. If it does, your solution is correct.

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.

Trigonometric Functions

Trigonometric functions, such as secant (sec) and tangent (tan), are fundamental in calculus, particularly in integration and differentiation. The secant function is defined as the reciprocal of the cosine function, while the tangent function is the ratio of sine to cosine. Understanding their derivatives and integrals is crucial for solving problems involving these functions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values. This theorem is essential for verifying the correctness of indefinite integrals by differentiating the result to check if it matches the original integrand.