~Common Integration Errors~



~In the initial stages of the integration process, the student is given the most popular forms that occur in the most basic problems.

~They include forms for sums and differences, power rule, basic exponential, basic trig, and the form that gives the natural log in the answer.

~NOTE: All of these forms contain u and du, where u is a function of the variable of integration (usually x, t, or θ). The differential of u, du, is computed by taking the derivative of u with respect to that variable multiplied by the differential of that variable.

~Error #1: Choosing the incorrect form to fit your integral. If you do this, you have no chance of solving your problem correctly. It's like trying to fit a full grown elephant into the back seat of a volts wagon beetle.

~Choosing the right integration formula to fit your problem is essential at the start.

~Error #2: Assuming you have chosen the correct form for your problem, you fail to locate the quantity which is represented by u in the formula.

~This quantity u occupies different positions depending on the form used. For example, in the power rule form, it's the expression which is raised to the power n. In the basic trig form, it's the argument (angle position) of the trig expression. In the basic exponential form, it's the power on e, just to name a few.

~If you choose u incorrectly, your problem cannot be integrated properly.

~Error #3: Assuming you have located the quantity u correctly, your computation of the differential of u, namely du is in error. To compute the proper du, simply take the derivative with respect to the variable of your problem then multiply the result by the differential of that variable. A wrong du will lead you to an incorrect adjustment in your problem (if needed), consequently, the wrong answer.

~Error #4: Assuming you have the correct u and du, you make the necessary adjustment (if needed) incorrectly.

~The only type of adjustments allowed are those where you multiply the integrand (the expression to be integrated) by a constant and divide the outside (far left) by the same constant. That way, you are not changing your problem.

~The most common error made here is multiplying and dividing by an expression containing the variable of the problem. No variable quantities can be used in this type of adjustment. That's a no-no.

~Error #5: Assuming you've gotten this far correctly without error, you now are ready to state your final answer to the integral by reading the form answer in the integration formula that was used. Many students do not state their answers correctly. The u in the form answer is the same u that was located previously. All parts of the du that you formulated previously will not appear in the answer, it simply gets reabsorbed back into the problem. Calculus is one of the few math courses where "things" disappear in the process of stating the answer.

~Error #6: Finally, the least serious of all mistakes made is forgetting the +C in your answer. For all indefinite integrals, you must include this in the answer. Definite integrals do not possess this problem. When dealing with definite integrals that have numerical limits, the final answer is just a number.

~NOTE: For definite integrals with numerial limits, the TI-83 or TT-84 calculators (or equivalent), can be used to get the final answer without doing much work. Since these types of definite integrals are just numbers, this number can be calculated as follows:

(1) code in the function in the integrand (expression to be integrated). This is located between the integral symbol & the differential on the far right.

(2) adjust the x window (if necessary) to span the limits of integration. If your x window does not include these limits, an error message will occur. If a sketch is desired, adjust the y window to include values of the integrand over your x window values. Values of the integrand can be viewed by going to the table (2nd graph) & entering various x values. Just make sure your table is set-up properly (go to 2nd window & make sure Indpnt: is set to ask and Depent: to auto. However, a sketch is not necessary to get the answer.

(3) go to 2nd trace (Calc) down to menu 7. Press enter. You will be prompted to enter the lower limit, enter. Then you will be prompted to enter the upper limit, enter. The answer will be displayed along with the sketch (if desired). However, most answers given are approximations and may not satisfy the testing situations of some teachers. Numerial answers involving e, π, or other irrational numbers will be approximated. To get these exact answers, follow the integration procedures correctly except for the constant. Instead, find the numerial value of the difference in your answer after the upper & lower limits have be substituted into your answer.(i.e., if a & b are the lower & upper limits of integration, resp. and F(x) is the answer without the constant, compute F(b)-F(a)).

~Conclusion: Integrating correctly is somewhat of a challenge for beginning students initially. Only practice and studying correctly done problems will increase your likelihood of solving them correctly.