Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫19 x2−4 ⅆx.
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.
∫19 x2−4 ⅆx
= ∫23secθtanθ dθ2 tanθ
The integral of secθ is derived in Table 6.2.10. The absolute value in line 3 is retained in line 4 because the argument of the logarithm is negative for θ∈π,3 π/2, which corresponds to x<−2/3
Evaluate the given integral
Control-drag the integral.
Press the Enter key.
Context Panel: Simplify≻Simplify
∫19 x2−4 ⅆx = 19⁢ln⁡x⁢9+9⁢x2−4⁢9= simplify 13⁢ln⁡3⁢x+9⁢x2−4
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≔∫19 x2−4 ⅆx:
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 cosine function is "simpler" than the secant.)
q2≔simplifyq1 assuming θ∷RealRange0,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.18(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, tanθ=129 x2−4. Note how the simplification isolates the additive constant of integration, −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=9⁢x2−4−3⁢x and proceeds as shown in Table 6.3.18(a).
Table 6.3.18(a) The substitution u=9⁢x2−4−3 x made by the Integration Methods tutor
Table 6.3.18(b) shows the result when the Change rule x=23secθ is imposed on the tutor.
Table 6.3.18(b) Integration Methods tutor after x=23secθ is imposed
The integrand has been simplified without any restriction on θ; hence, the absolute value of secθ appears. The only way to proceed in the tutor is via the Rewrite rule whereby secθ is simply replaced with secθ, at which point the stepwise code will re-derive the antiderivative of secθ as per 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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