Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫9 x2−4x2ⅆx.
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.
= ∫2 tanθ23secθtanθ dθ23secθ2
=3 lnsecθ+tanθ−3 sinθ
=3 ln32x+9 x2−42−39 x2−43 x
=3 ln32x+9 x2−42−9 x2−4x
The absolute value in line 5 is retained in line 6 because the argument of the logarithm is negative for θ∈π,3 π/2. Figure 6.3.3 is the basis for the substitution sinθ=9 x2−43 x.
Evaluate the given integral
Control-drag the integral and press the Enter key.
Context Panel: Simplify≻Simplify
Context Panel: Expand≻Expand
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.
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 sines and cosines are "simpler" than tangents and secants.)
q2≔simplifyq1 assuming θ∷RealRange0,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.24(b), below.
Revert the change of variables by applying the substitution θ=arcsec3 x/2.
simplifyevalq3,θ=arcsec32x assuming x≥2/3
From Figure 6.3.3, sinθ=13 x9 x2−4, and tanθ=129 x2−4. This solution differs from the previous one by an additive constant of integration.
The stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution u=9 x2−4−3 x and proceeds as shown in Table 6.3.24(a).
Table 6.3.24(a) The substitution u=9 x2−4−3 x made by the Integration Methods tutor
The rational function that results from Maple's change of variables yields to the algebraic technique of partial fraction decomposition, a technique that will be studied in Section 6.4. The integration of u/u2+4 is handled by the change of variables v=u2+4, but it would also yield to the substitution u=2 tanθ.
Table 6.3.24(b) shows the result when the Change rule x=23secθ is imposed on the tutor.
Table 6.3.24(b) Integration Methods tutor after x=23secθ is imposed
It takes the Rewrite rule to apply the trig identity sin2θ=1−cos2θ, and to change 1/cosθ to secθ. The integration of secθ is detailed in Table 6.2.10.
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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