Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫9 x2−4xⅆx.
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 dθ23secθ
Passage from line 2 to line 3 is via the trig identity tan2θ=sec2θ−1. The antiderivative of sec2θ is tanθ because the derivative of tanθ is sec2θ.
Finally, note that here, θ=arcsec32x or θ=arctan9 x2−42.
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫9 x2−4xⅆx = 9⁢x2−4+2⁢arctan⁡29⁢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.
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 tangents. )
q2≔simplifyq1 assuming θ∷RealRange0,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.21(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.
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 6.3.21(a). The change of variables selected by the tutor leads to the rational function u2/u2+4, which immediately becomes 1−4/u2+4 by long division. The antiderivative of 1/u2+4 can be found by coercing this fraction into the form 1/z2+1, so that arctanz results. The tutor, however, makes the change of variables u=2 tanv, which converts the integrand to 1/2.
∫9⁢x2−4xⅆx=∫1−4u2+4ⅆuchange,9⁢x2−4=u2,u=∫1ⅆu−4⁢∫1u2+4ⅆusum=u−4⁢∫1u2+4ⅆuconstant=u−4⁢∫12 dvchange,u=2⁢tan⁡v,v=u−2 vconstant=u−2⁢arctan⁡u2revert=9⁢x2−4−2⁢arctan⁡9⁢x2−42revert
Table 6.3.21(a) The substitution u2=9 x2−4 made by the Integration Methods tutor
Table 6.3.21(b) shows the result when the Change rule x=23secθ is imposed on the tutor.
Table 6.3.21(b) Integration Methods tutor after x=23secθ is imposed
After making the initial change of variables, the integrand is actually 2 tan2θ, which becomes 2 u2/u2+1. As in Table 6.3.21(a), Maple immediately writes this rational function as 21−1/u2+1. The further change of variables u=tanv changes ∫duu2+1 to ∫dv. Here, it is well to note that the integration rules are names for the properties or expressions in the integrand. Therefore, the arctan rule does not apply because the arctangent arises not in the integrand, but only after the integral is evaluated. That's why the additional change of variables is needed in both Tables 6.3.21(a) and 6.3.21(b).
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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