1. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Consider a continuous function f. If F is an antiderivative for f on [a, b], then If C is a constant, then Thus we can replace (1) by writing If we have no particular interest in the interval [a, b] but wish instead to emphasize that F is an antiderivative for f, which on open intervals simply means that F´ = f, then we omit the a and the b and simply write Antiderivatives expressed in this manner are called indefinite integrals. The constant C is called the constant of integration; it is an arbitrary constant and we can assign to it any value we choose. Each value of C gives a particular antiderivative, and each antiderivative is obtained from a particular value of C.

2. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals

3. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals The linearity properties of definite integrals also hold for indefinite integrals. Example 1 Calculate Solution

4. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 2 Find f given that

5. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 3 Find f given that

6. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Application to Motion Example 4 An object moves along a coordinate line with velocity v(t) = 2 − 3t + t 2 units per second. Its initial position (position at time t = 0) is 2 units to the right of the origin. Find the position of the object 4 seconds later. Solution

7. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 5 An object moves along the x-axis with acceleration a(t) = 2t - 2 units per second per second. Its initial position (position at time t = 0) is 5 units to the right of the origin. One second later the object is moving left at the rate of 4 units per second. (a)Find the position of the object at time t= 4 seconds (b)How far does the object travel during these 4 seconds?

8. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 6 Find the equation of motion for an object that moves along a straight line with constant acceleration a from an initial position x0 with initial velocity v0.

9. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution

10. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 1 Calculate

11. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 2 Calculate

12. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 3 Calculate

13. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 4 Calculate Solution

14. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 5 Calculate

15. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 6 Calculate

16. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 7. Calculate

17. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 8 Evaluate

18. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 9 Evaluate

19. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 10 Evaluate

20. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 11 Calculate

21. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 12. Evaluate

22. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution The Definite Integral