Example 1: # 10 (p. 385)

Here are the steps needed to use the Integration maplet to evaluate the integral

0. Launch the Integration  maplet (see above)

1. Enter the problem by filling in the fields at the top of the maplet window:

     Function : x^2/sqrt(16-x^2)  

     Variable : x

    The from  and to  fields are used to specify the lower and upper limits of integration for a definite integral.

2. Click the Start  button along the bottom row of the maplet.

3. The first step in this problem is to rationalize the integrand with the substitution: x=4*sin(t) . To implement this step, enter x=4*sin(t)  in the field to the right of the rewrite  and change  buttons. Then click the change  button. The result should be

4. To integrate  it is necessary to use the double-angle identity: . To complete this step, enter sin(t)^2=(1-cos(2*t))/2  in the field to the right of the rewrite  and change  buttons. Now click the rewrite  button.

5. While factoring the 8 from both terms of the integrand would be a reasonable next step, the Integration maplet does not permit it. (Click the Constant Multiple  button to see what happens.) Another possible step is to separate the integral into two separate integrals. To do this click the Difference  (or Sum ) button.

6. The first integral can now be evaluated using the Constant rule (click the Constant button).

7. For the remaining term, implement the change of variables  by entering 2*t=u  in the field to the right of the rewrite and change  buttons then click change .

8. Click Constant Multiple  to bring the factor of  outside the integral.

9. Click Select a Function . From the list that appears, click cos . Now click the Apply button.

10. There should be no integrals left in the problem. All that remains is to undo the change of variables. To do this, click the Revert  button - twice.

11. The final answer should be

Of course, you would add a constant of integration.

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Example 2: # 18 (p. 385)

0. To evaluate

begin by entering the problem and clicking Start.

1. To complete the square in the expression under the radical, use the Integration maplet to rewrite    as .

2. Make the change  of variables . (Note the distinction between rewrite  and change .)

3. To rationalize the integrand, use the change  of variables .

4. The resulting integral can be evaluated with an appropriate Function Rule.

5. Use the revert button as many times as necessary to undo the substitutions.

The final result, without the constant of integration, should be

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Example 3: # 17 (p. 390)

To evaluate the definite integral

enter the integrand and variable of integration as usual. In addition, fill in the from  field with 1  and the to  field with exp(1) . Integration by parts is the method of choice for this problem. Because there is no easy antiderivative for the natural logarithm, the only real choice is to choose  and . Then

    =   .

Enter the value of  ( ln(t) ) in the field next to f(x)=  and the value of  ( 2/3*t^(3/2) ) in the field next to g(x)= . Now, click the Int by Parts  button. Note that the u*v term is automatically evaluated at the upper and lower limits of integration. To finish the problem apply the Constant Multiple  rule followed by the Power  rule.

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Example 4: # 14 (p. 390)

This problem, , is similar to Example 3 in that there is no easy antiderivative for the arctan function. Thus, choose u=arctan(5*x) and dv=dx. Then  = . Enter this information in the f(x)=  and g(x)=  fields and click Int by Parts . The integral produced by this step involves an integral that can be evaluated in a number of different ways. Seeing that the denominator  =  when . Make this change  of variable. The next step is the ever-present Constant Multiple  rule. To conclude, select the function tan  (or simply type tan  in the Enter a Function  field), then click Apply , then Revert  to the original variable. (Note that the maplet uses ln in this result.)

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Note: Using Understood Rules

If you are like me, you are tired of having to click the Constant Multiple , Sum , and Difference  buttons. There is a way to tell the Integration  maplet to automatically apply these, or most other, rules. In the menu bar, click on Understood Rules , then click on Constant Multiple . Repeat for Sum Rule  and Difference Rule .

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Example 5: # 46 (p. 390)

This is one of the classical integrals found in every calculus course. Because differentiation or antidifferentiation of the exponential yields another exponential and derivatives and antiderivatives of sin and cos alternate, there is no real hope that integration by parts will ``simplify'' the integrand. But, the periodic nature of the trigonometric functions means we might end up where we started. Let's see this, then discuss how to complete the problem.

To start, enter the integrand and the variable of integration. There are two choices for the f(x)=  and g(x)=  fields. If you choose  and , then . If you choose  and , then   . Enter one of these choices in the f(x)=  and g(x)=  fields and click Int by Parts . The integral that this produces will have an integrand that includes .

Now, apply integration by parts with the exponential in the same place as in the first integration by parts. If you do this correctly the second integration by parts will involve the original integral -- with a coefficient different from 1. (If you did reversed the choices of u and dv between the two applications of integration by parts you will end up --- after simplification --- with the original integral.)

You now have an equation that looks something like

where . It is easy to solve this equation for :  . The Solve  button will do this calculation within the Integration  maplet. The final result should be

   .

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Note: Applying Partial Fractions

For these examples it is helpful to let Maple find the partial fractions decomposition. On first glance this does not appear possible. But, if you will type partialfractions  (all one word, all lower case) in the field next to Enter a Function  and then click Apply , you will see the result of applying the partial fractions decomposition to the integrand.

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Example 6: # 6 (p. 397)

Consider the integral

The denominator factors as . This means we expect a partial fraction decomposition of the integrand to have the form B[1]/(x-4) + B[2]/(x+3). To complete this decomposition, type partialfractions  in the Enter a Function  field and click Apply . (Note that the Constant Multiple and Sum, or Difference, rules have been applied automatically.) The remaining integrals require separate change of variables. It is recommended that you work from left to right and completely evaluate each integral before beginning the next.

The final result from the Integration maplet should have the form . How could you rewrite this as a single logarithm?

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Example 7: # 22 (p. 397)

Consider the integral

   .

The integrand is a rational function and . Enter this problem in the maplet and Apply  the partialfractions  function to this problem. Note that the two integrals have denominators of  and  and the numerator in each case is a constant. The same change of variable can be used to complete the evaluation of these integrals. As suggested above, work left to right and complete each integral before beginning the next.

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Example 8: # 35 (p. 397)

In this example the denominator involves a repeated (ireducible) quadratic factor. In general, a term  in the denominator will lead to  terms in the partial fraction decomposition. These  terms will be of the form    for , 2, ..., .

In # 35 the denominator contains . This means we should expect to see two terms in the partial fraction decomposition of the integrand; one with denominator , the other with denominator .

Enter this problem in the Integration  maplet and click Start . The first step is to find the partial fraction decomposition (see Note 2, above). Notice that there are two terms in the partial fraction decomposition. These integrals can be completed by applying a change of variable and the Power rule. Is your final answer

    ?

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