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.

*PROBLEM 1 :*Integrate .Click HERE to see a detailed solution to problem 1.

*PROBLEM 2 :*Integrate .Click HERE to see a detailed solution to problem 2.

*PROBLEM 3 :*Integrate .Click HERE to see a detailed solution to problem 3.

*PROBLEM 4 :*Integrate .Click HERE to see a detailed solution to problem 4.

*PROBLEM 5 :*Integrate .Click HERE to see a detailed solution to problem 5.

*PROBLEM 6 :*Integrate .Click HERE to see a detailed solution to problem 6.

*PROBLEM 7 :*Integrate .Click HERE to see a detailed solution to problem 7.

*PROBLEM 8 :*Integrate .Click HERE to see a detailed solution to problem 8.

*PROBLEM 9 :*Integrate .Click HERE to see a detailed solution to problem 9.

*PROBLEM 10 :*Integrate .Click HERE to see a detailed solution to problem 10.

*PROBLEM 11 :*Integrate .Click HERE to see a detailed solution to problem 11.

*PROBLEM 12 :*Integrate .Click HERE to see a detailed solution to problem 12.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :