Chapter 6: Techniques of Integration
Section 6.1: Integration by Parts
Using the technique of integration by parts, evaluate the definite integral ∫01tan−1x ⅆx.
The antiderivative of tan−1x is found much like the antiderivative of lnx was found in Example 6.1.2. Use parts integration with u=tan−1x and dv=dx so that u′=1/1+x2 and v=x. The complete calculation appears below where the limits of integration are included as per Table 6.1.1. The first two steps are a straightforward application of integration by parts that results in a new integral that yields to a substitution as per the annotation to the right of the step. Limits of integration are left in terms of x in anticipation of a return to the variable x after the appropriate antiderivative is found.
=x tan−1xx=0x=1−∫01x1+x2 ⅆx
(The substitution y=1+x2 is made so that dy=2 x dx and x dx=dy/2. The limits are left in terms of x.)
(The antiderivative is found in terms of y.)
(Revert from y to 1+x2.)
Evaluation of the integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫01tan−1x ⅆx = 14⁢π−12⁢ln⁡2
Annotated stepwise solution
Tools≻Load Package: Student Calculus1
Control-drag the definite integral.
Context Panel: Student Calculus1≻All Solution Steps
∫01tan−1x ⅆx→show solution stepsIntegration Steps∫01arctan⁡xⅆx▫1. Apply integration by Parts◦Recall the definition of the Parts rule∫uⅆv=u⁢v−∫vⅆu◦First partu=arctan⁡x◦Second part=1◦Differentiate first part=ⅆⅆxarctan⁡x=1x2+1◦Integrate second partv=∫1ⅆxv=x∫arctan⁡xⅆx=arctan⁡x⁢x−∫xx2+1ⅆxThis gives:π4−∫01xx2+1ⅆx▫2. Apply a change of variables to rewrite the integral in terms of u◦Let u beu=x2+1◦Isolate equation for xx=u−1◦Differentiate both sides=2⁢u−1◦Substitute the values for x and dx back into the original∫01xx2+1ⅆx=∫1212⁢uⅆuThis gives:π4−∫1212⁢uⅆu▫3. Apply the constant multiple rule to the term ∫12⁢uⅆu◦Recall the definition of the constant multiple rule∫⁢f⁡uⅆu=⁢∫f⁡uⅆu◦This means:∫12⁢uⅆu=∫1uⅆu2We can rewrite the integral as:π4−∫121uⅆu2▫4. Apply the reciprocal rule to the term ∫1uⅆu◦Recall the definition of the reciprocal rule∫1uⅆu=ln⁡u◦Apply limits of definite integral−We can rewrite the integral as:π4−ln⁡22
While Maple "understands" the notation tan−1x as input, the notation arctanx is used as output. For the change of variables, Maple uses u=1+x2 and also changes the limits of integration so that x=0→u=1 and x=1→u=2.
This annotated stepwise solution can also be obtained interactively with the
tutor. After the Parts rule is applied, the Change rule must be applied to effect the change of variables in the new integral. Pressing the Close button will write an annotated solution to the worksheet.
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.
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