Chapter 6: Techniques of Integration
Section 6.1: Integration by Parts
Using the technique of integration by parts, evaluate ∫lnx ⅆx.
The indefinite integral ∫lnx ⅆx can be evaluated by parts integration if u=lnx,dv=dx, so that v=x and u′=1/x. The complete calculation is as follows.
= x lnx−∫x⋅1/x ⅆx
= x lnx−∫1 ⅆx
= x lnx−x
Annotated stepwise solution
Tools≻Load Package: Student Calculus 1
Expression palette: Indefinite-integral template
Context Panel: Student Calculus1≻All Solution Steps
∫lnx ⅆx→show solution stepsIntegration Steps∫ln⁡xⅆx▫1. Apply integration by Parts◦Recall the definition of the Parts rule∫uⅆv=v⁢u−∫vⅆu◦First partu=ln⁡x◦Second part=1◦Differentiate first part=ⅆⅆxln⁡x=1x◦Integrate second partv=∫1ⅆxv=x∫ln⁡xⅆx=ln⁡x⁢x−∫1ⅆxThis gives:ln⁡x⁢x−∫1ⅆx▫2. Apply the constant rule to the term ∫1ⅆx◦Recall the definition of the constant rule∫Cⅆx=C⁢x◦This means∫1ⅆx=xWe can now rewrite the integral as:ln⁡x⁢x−x
Maple applies parts integration with u=lnx and v=x, as per the mathematical solution given above.
tutor provides for this same solution interactively. Press the Parts button, enter fx=lnx and gx=x for u and v, respectively, and complete the integration by selecting the Constant rule. When the tutor is closed, the same annotated stepwise solution is written to the worksheet. This exploration is left to the reader.
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.
The Parts command in the IntegrationTools package only requires that u be declared. Table 6.1.2(a) shows this command being accessed through the
Tools≻Tasks≻Browse: Calculus - Integral≻Methods of Integration≻Parts
Integration by Parts
Enter the integral ∫u dv:
Execute integration by parts:
Table 6.1.2(a) Integration by Parts task template
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