1. CHAPTER 6: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING SECTION 6.2: ANTIDIFFERENTIATION BY SUBSTITUTION AP CALCULUS AB

2. What you’ll learn about Indefinite Integrals Leibniz Notation and Antiderivatives Substitution in Indefinite Integrals Substitution in Definite Integrals … and why Antidifferentiation techniques were historically crucial for applying the results of calculus.

3. Section 6.2 – Antidifferentiation by Substitution Definition: The set of all antiderivatives of a function f(x) is the indefinite integral of f with respect to x and is denoted by

4. Section 6.2 – Antidifferentiation by Substitution is read “The indefinite integral of f with respect to x is F(x) + C.” Example: constant of integration integral sign integrand variable of integration

5. Section 6.2 – Antidifferentiation by Substitution Integral Formulas: Indefinite IntegralCorresponding Derivative Formula 1. 2. 3.

6. Section 6.2 – Antidifferentiation by Substitution More Integral Formulas: Indefinite IntegralCorresponding Derivative Formula 4. 5.

7. Section 6.2 – Antidifferentiation by Substitution More Integral Formulas: Indefinite IntegralCorresponding Derivative Formula 6. 7. 8. 9.

8. Trigonometric Formulas

9. Section 6.2 – Antidifferentiation by Substitution Properties of Indefinite Integrals: Let k be a real number. 1. Constant multiple rule: 2. Sum and Difference Rule:

10. Section 6.2 – Antidifferentiation by Substitution Example: C’s can be combined into one big C at the end.

11. Example Evaluating an Indefinite Integral

12. Section 6.2 – Antidifferentiation by Substitution Remember the Chain Rule for Derivatives: By reversing this derivative formula, we obtain the integral formula

13. Section 6.2 Antidifferentiation by Substitution Power Rule for Integration If u is any differentiable function of x, then

14. Exponential and Logarithmic Formulas

15. Section 6.2 – Antidifferentiation by Substitution A change of variable can often turn an unfamiliar integral into one that we can evaluate. The method for doing this is called the substitution method of integration.

16. Section 6.2 – Antidifferentiation by Substitution Example:

17. Rules For Substitution-- A mnemonic help: L - Logarithmic functions: ln x, log b x, etc.Logarithmic functions I - Inverse trigonometric functions: arctan x, arcsec x, etc.Inverse trigonometric functions A - Algebraic functions: x 2, 3x 50, etc.Algebraic functions T - Trigonometric functions: sin x, tan x, etc.Trigonometric functions E - Exponential functions: e x, 19 x, etc.Exponential functions The function which is to be dv is whichever comes last in the list: functions lower on the list have easier antiderivatives than the functions above them.antiderivatives The rule is sometimes written as "DETAIL" where D stands for dv.

18. To demonstrate the LIATE rule, consider the integral: Following the LIATE rule, u = x and dv = cos x dx, hence du = dx and v = sin x, which makes the integral become which equals

19. Section 6.2 – Antidifferentiation by Substitution Example:

20. Example Paying Attention to the Differential

21. Example Using Substitution

23. Example Setting Up a Substitution with a Trigonometric Identity Hint: let u = cos x and –du = sinxdx

24. Section 6.2 – Antidifferentiation by Substitution Substitution in Definite Integrals Substitute and integrate with respect to u from

25. Section 6.2 – Antidifferentiation by Substitution Ex:

26. Example Evaluating a Definite Integral by Substitution

27. Section 6.2 – Antidifferentiation by Substitution You try:

28. Section 6.2 – Antidifferentiation by Substitution You try:

29. Section 6.2 – Antidifferentiation by Substitution You try:

30. Section 6.2 – Antidifferentiation by Substitution You try: