### 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 :

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.