Substitution Lesson 7.2

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

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)

Integration by Substitution Now we have Then we use the general power rule for integrals Finally substitute u = x2 + 1 back in

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

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 for the 4 in the du

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

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

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

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

Indefinite Integral of eu Try this: What is the u? the du? u = x4 du = 4x3 dx Rewrite, adjust for the factor of 4 in the du

Practice Try these

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

Assignment Lesson 7.2A Page 449 Exercises 3 – 41 odd Lesson 7.2B Exercises 39 – 44 all