The following problems involve the integration of rational functions, resulting in logarithmic or inverse tangent functions. We will assume knowledge of the following well-known differentiation formulas :

,

where
represents the natural (base *e*) logarithm of *x* and

,

where represents the inverse function of . These formulas lead immediately to the following indefinite integrals :

- ,
- .

Because the integral

,

where *a* is any positive constant, appears frequently 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 rewriting the function. Then

.

Now let

*u*= *x*/*a*

so that

*du* = (1/*a*) *dx* ,

or

(*a*)*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.

Recall also the method of algebraically "completing the square". For example to complete the square with *x*^{2}+6*x* we divide 6 by 2, square it getting 9, add and subtract 9, so that

*x*^{2}+6*x* = (*x*^{2}+6*x*+9) - 9 = (*x*+3)^{2} - 9 .

To complete the square with *x*^{2}+*Bx* , divide *B* by 2, square it getting
(*B*/2)^{2}=*B*^{2}/4, add and subtract *B*^{2}/4, so that

*x*^{2}+*Bx* = (*x*^{2}+*Bx*+*B*^{2}/4) - *B*^{2}/4 = (*x*+(*B*/2))^{2} - *B*^{2}/4 .

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

.

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. There is a very high emphasis on arithmetic and algebra in this problem set. AVOID MAKING THE FOLLOWING VERY COMMON ALGEBRA MISTAKE :

THIS IS FALSE :
.

Convince yourself that this is generally INCORRECT by plugging numbers in for *A*, *B*, and *C* (Try *A*=4, *B*=2, and *C*=2 .).

*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.

*PROBLEM 13 :*Integrate .Click HERE to see a detailed solution to problem 13.

*PROBLEM 14 :*Integrate .Click HERE to see a detailed solution to problem 14.

*PROBLEM 15 :*Integrate .Click HERE to see a detailed solution to problem 15.

*PROBLEM 16 :*Integrate .Click HERE to see a detailed solution to problem 16.

*PROBLEM 17 :*Integrate .Click HERE to see a detailed solution to problem 17.

*PROBLEM 18 :*Integrate .Click HERE to see a detailed solution to problem 18.

*PROBLEM 19 :*Integrate .Click HERE to see a detailed solution to problem 19.

*PROBLEM 20 :*Integrate .Click HERE to see a detailed solution to problem 20.

*PROBLEM 21 :*Integrate .Click HERE to see a detailed solution to problem 21.

*PROBLEM 22 :*Integrate .Click HERE to see a detailed solution to problem 22.

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