Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x29 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.
∫x29 x2−4 ⅆx
= ∫23secθ223secθtanθ dθ2 tanθ
=227(32x9 x2−42+ln(32x+9 x2−42))
=x189 x2−4+227ln(32x+9 x2−42)
The integral of sec3θ evaluated in line 3 is derived in Example 6.2.5. 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.
Evaluate the given integral
Control-drag the integral and press the Enter key.
Context Panel: Simplify≻Simplify
∫x29 x2−4 ⅆx
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≔∫x29 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 6.3.23(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, cosθ=23 x, 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 6.3.23(a).
The change of variables selected by the tutor leads to a multiple of u2+42/u3, provided u≥2, and this corresponds to x≤−2/3. Since the tutor does not have provision for routine simplifications, it takes several steps, including invocation of the Rewrite rule to massage the expression into a form where the power rule of integration applies.
Table 6.3.23(a) The substitution u=9 x2−4−3 x made by the Integration Methods tutor
Table 6.3.23(b) shows the result when the Change rule x=23secθ is imposed on the tutor.
Table 6.3.23(b) Integration Methods tutor after x=23secθ is imposed
It takes the Rewrite rule to remove the absolute value in the integrand, and without such, the tutor can go no further. The antiderivative of sec3θ is derived in Example 6.2.5.
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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