Chapter 6: Techniques of Integration
Section 6.1: Integration by Parts
Example 6.1.1
Using the technique of integration by parts, evaluate ∫x sinx ⅆx.
Solution
Mathematical Solution
When applying ∫u dv=u v−∫v ⅆu, the choice u=x,dv=sinx dx is optimal because u′=1 and v=−cosx. The following steps then complete the calculation.
∫x sinx ⅆx
=x −cosx−∫1⋅−cosx ⅆx
=−x cosx+∫cosx ⅆx
=−x cosx+sinx
(The choice u=sinx,dv=x dx fails because v=x2/2 then requires the more difficult integration of x2 cosx.)
Maple Solutions
An annotated stepwise solution can be obtained with the Context Panel system.
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Expression palette: Indefinite-integral template
Context Panel: Student Calculus1≻All Solution Steps
∫x sinx ⅆx→show solution stepsIntegration Steps∫xsinxⅆx▫1. Apply integration by Parts◦Recall the definition of the Parts rule∫uⅆv=vu−∫vⅆu◦First partu=x◦Second partdv=sinx◦Differentiate first partdu=ⅆⅆxxdu=1◦Integrate second partv=∫sinxⅆxv=−cosx∫xsinxⅆx=−xcosx−∫−cosxⅆxThis gives:−xcosx−∫−cosxⅆx▫2. Apply the constant multiple rule to the term ∫−cosxⅆx◦Recall the definition of the constant multiple rule∫Cfxⅆx=C∫fxⅆx◦This means:∫−cosxⅆx=−∫cosxⅆxWe can rewrite the integral as:−xcosx+∫cosxⅆx▫3. Evaluate the integral of cos(x)◦Recall the definition of the cos rule∫cosxⅆx=sinxThis gives:−xcosx+sinx
In the first step, Maple indicates that integration by parts has been used, and shows u=x and v=−cosx.
The tutor provides for the interactive application of parts integration. Figure 6.1.1(a) shows the tutor after the Parts button has been pressed and the fields for fx (i.e., u) and gx (i.e., v) have been entered.
Figure 6.1.1(a) Application of the Integration Methods tutor
Clicking the Apply button will put the tutor in the state shown in Figure 6.1.1(b), from which the remaining steps can be deduced from the annotated stepwise solution given above.
Figure 6.1.1(b) Integration Methods tutor after parts integration invoked
Note that an annotated stepwise solution is available via the Context Panel with the "All Solution Steps" option.
The rules of integration can also be applied via the Context Panel, as per the figure to the right.
Maple has one other tool for applying integration by parts.
Copy/paste the inert integral. Context Panel: Assign to a Name≻q
∫x sinx ⅆx→assign to a nameq
Apply the Parts command from the IntegrationTools package.
IntegrationTools:-Partsq,x
−xcosx−∫−cosxⅆx
Note that the Parts command in the IntegrationTools package requires only that u be known. The command then deduces dv and v from the integrand. This command can also be accessed through the task template shown in Table 6.1.1(a).
Tools≻Tasks≻Browse: Calculus - Integral≻Methods of Integration≻Parts
Integration by Parts
Enter the integral ∫u dv:
Intx sinx, x
∫xsinxⅆx
Declare u:
x
Execute integration by parts:
IntegrationToolsParts,
−cosxx−∫−cosxⅆx
Table 6.1.1(a) Integration by Parts task template
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