### THE INTEGRATION OF EXPONENTIAL FUNCTIONS

The following problems involve the integration of exponential functions. We will assume knowledge of the following well-known differentiation formulas : ,

where , and ,

where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a . These formulas lead immediately to the following indefinite integrals :

1. 2. As you do the following problems, remember these three general rules for integration : ,

where n is any constant not equal to -1, ,

where k is any constant, and .

Because the integral ,

where k is any nonzero constant, appears so often in the following set of problems, we will find a formula for it now using u-substitution so that we don't have to do this simple process each time. Begin by letting

u=kx

so that

du = k dx ,

or

(1/k)du = dx .

Now substitute into the original problem, replacing all forms of x, and getting    .

We now have the following variation of formula 1.) :

3. .

The following often-forgotten, misused, and unpopular rules for exponents will also be helpful : and .

Most of the following problems are average. A few are challenging. Knowledge of the method of u-substitution will be required on many of the problems. Make precise use of the differential notation dx and du and always be careful when arithmetically and algebraically simplifying expressions.

• PROBLEM 1 : Integrate .

• PROBLEM 2 : Integrate .

• PROBLEM 3 : Integrate .

• PROBLEM 4 : Integrate .

• PROBLEM 5 : Integrate .

• PROBLEM 6 : Integrate .

• PROBLEM 7 : Integrate .

• PROBLEM 8 : Integrate .

• PROBLEM 9 : Integrate .

• PROBLEM 10 : Integrate .

• PROBLEM 11 : Integrate .

• PROBLEM 12 : Integrate .