Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Example 6.3.17
Evaluate the indefinite integral ∫9 x2−4ⅆx.
Solution
Mathematical Solution
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.
∫9 x2−4ⅆx
= ∫2 tanθ23secθtanθ dθ
=43∫tan2θsecθ dθ
=43∫sec2θ−1secθ dθ
=43∫sec3θ dθ−∫secθ dθ
=43(secθtanθ+lnsecθ+tanθ/2−ln(secθ+tanθ))
23secθtanθ−lnsecθ+tanθ
=2332x9 x2−42−23ln32x+9 x2−42
=x29 x2−4−23ln32x+9 x2−42
Line 3 is obtained by applying the trig identity tan2θ=sec2θ−1. The integral of sec3θ evaluated in line 5 is derived in Example 6.2.5. The integral of secθ is derived in Table 6.2.10. The absolute values in line 5 are dropped in line 6 because the argument of the logarithm is positive for the θ-interval defined in Table 6.3.1. (The astute reader will note that the trig integral evaluated in this example is the same as the one in Example 6.3.15.)
Maple Solution
Evaluate the given integral
Control-drag the integral and press the Enter key.
Context Panel: Simplify≻Simplify
12x9x2−4−29lnx9+9x2−49
= simplify
12x9x2−4−23ln3x+9x2−4
A stepwise solution that uses top-level commands except for one application of the Change command from the IntegrationTools package:
Initialization
Install the IntegrationTools package.
withIntegrationTools:
Let Q be the name of the given integral.
Q≔∫9 x2−4ⅆx:
Change variables as per Table 6.3.1
Use the Change command to apply the change of variables x=23secθ.
q1≔ChangeQ,x=23secθ
∫234secθ2−4secθtanθⅆθ
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.)
q2≔simplifyq1 assuming θ∷RealRange0,π2
43∫sinθ2cosθ3ⅆθ
Use the value command to evaluate the integral, or follow the approach in Table 6.3.17(b), below.
q3≔valueq2
23sinθ3cosθ2+23sinθ−23lnsecθ+tanθ
Revert the change of variables by applying the substitution θ=arcsec3 x/2.
simplifyevalq3,θ=arcsec32x assuming x∷RealRange0,π/2
12x9x2−4−23ln3x+9x2−4+23ln2
From Figure 6.3.3, sinθ=13 x9 x2−4, cosθ=23 x, and tanθ=129 x2−4. Note how the simplification isolates the additive constant of integration, 2 ln2/3.
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=9x2−4−3x and proceeds as shown in Table 6.3.17(a).
∫9x2−4ⅆx=−∫uⅆu12+∫8u2−16u3ⅆu12change,u=9x2−4−3x,u=−u224+∫8u2−16u3ⅆu12power=−u224+2∫1uⅆu3−4∫1u3ⅆu3rewrite,8u2−16u3=8u−16u3
Table 6.3.17(a) The substitution u2=9+4/x2 made by the Integration Methods tutor
Note that the solution in Table 6.3.17(a) is not complete - the antiderivatives have not been obtained and the Revert rule has not been applied. Moreover, the Rewrite rule has to be used where, at top level, the expand command would have sufficed.
Table 6.3.17(b) shows the result when the Change rule x=23secθ is imposed on the tutor.
∫9x2−4ⅆx=2∫4secθ2−4secθtanθⅆθ3change,x=2secθ3=4∫u2u2+1ⅆu3change,u=tanθ,u
Table 6.3.17(b) Integration Methods tutor after x=23secθ is imposed
To put the integrand into the form of a multiple of secθtan2θ, the Rewrite rule would have to be applied. Maple's stepwise code instead applied the additional change of variable, u=tanθ, which actually does lead to a solution because the problem has now become the equivalent of Example 6.3.15.
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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