Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x 9 x2−4ⅆx.
The substitution x=23secθ means dx=23secθtanθ dθ, and turns hx into 2 tanθ. From Figure 6.3.3, tanθ=129 x2−4. Hence, the evaluation of the given integral proceeds as follows.
∫x 9 x2−4ⅆx
= ∫23secθ2 tanθ23secθtanθ dθ
=9 x2−4279 x2−4
Evaluate the given integral
Control-drag the integral.
Press the Enter key.
Context Panel: Simplify≻Simplify
∫x 9 x2−4ⅆx = 127⁢3⁢x−2⁢3⁢x+2⁢9⁢x2−4= simplify 127⁢9⁢x2−43/2
A stepwise solution that uses top-level commands except for one application of the Change command from the IntegrationTools package:
Install the IntegrationTools package.
Let Q be the name of the given integral.
Q≔∫x 9 x2−4ⅆx:
Change variables as per Table 6.3.1
Use the Change command to apply the change of variables x=23secθ.
Simplify the radical to 2 tanθ. Note the restriction imposed on θ.
(Maple believes that the sine and cosine functions are "simpler" than secants and tangents. Putting the integrand into the form sec2θtan2θ requires two separate substitution steps that in aggregate are more work than simply typing in the desired form.)
q2≔simplifyq1 assuming θ∷RealRange0,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.19(b), below.
Revert the change of variables by applying the substitution θ=arcsec3 x/2.
simplifyevalq3,θ=arcsec32x assuming x∷RealRange0,π/2
From Figure 6.3.3, sinθ=13 x9 x2−4, and cosθ=23 x.
The stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution u2=9 x2−4 and proceeds as shown in Table 6.3.19(a).
Table 6.3.19(a) The substitution u2=9 x2−4 made by the Integration Methods tutor
Table 6.3.19(b) shows the result when the Change rule x=23secθ is imposed on the tutor.
Table 6.3.19(b) Integration Methods tutor after x=23secθ is imposed
To put the integrand into the form of a multiple of sec2θtan2θ, the Rewrite rule would have to be applied. Maple's stepwise code instead applied the additional change of variable, u=tanθ, which is the appropriate substitution for integrating sec2θtan2θ.
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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