Substitution

1. Substitution Lesson 7.2

2. Review • Recall the chain rule for derivatives • We can use the concept in reverse • To find the antiderivatives or integrals of complicated formulas • We look for integrands that fit the right side of the chain rule

3. Strategy • We look for an expression that can be the "inside" function • We substitute u = g(x) • We also determine what is du or g'(x)

4. Integration by Substitution • Now we have • Then we use the general power rule for integrals • Finally substituteu = x2 + 1 back in

5. Substitution Method We seek the following situations where we can substitute u in as the "inner" function • Let u represent the quantity under a root or raised to a power • Let u represent the exponent on e • Let u represent the quantity in the denominator

6. Example • Consider the problem of taking the integral of • Strategy … substitute u = 4x – 6 • What is the derivative of u with respect to x? • Now we make the substitution • The ¼ adjusts forthe 4 in the du

7. Substitution • The resulting integral is much simpler • Now we reverse the substitution and simplify

8. Try Another • What will we substitute … u = ? • What is the du ? • Now rewrite the integral and proceed

9. How About Another? • Consider u = ? du = ? • u = x2 + 5 du = 2x dx • Problem … • 2x is not a constant • Cannot adjust the integral with a constant coefficient • Substitution will not work for this integral

10. Indefinite Integral of u -1 • If it looked like thiswe could do it • u = x2 + 5 du = 2x dx • Then use rule for integral of u -1 • Final result:

11. Indefinite Integral of eu • Try this: • What is the u? the du? • u = x4du = 4x3 dx • Rewrite, adjust for the factor of 4 in the du

12. Practice • Try these

13. Application • We are told that a certain bacteria population is increasing a rate of • What is the increase in the population during the first 8 hours

14. Assignment • Lesson 7.2A • Page 449 • Exercises 3 – 41 odd • Lesson 7.2B • Page 450 • Exercises 39 – 44 all