Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x 4+9 x2ⅆx.
The substitution x=23tanθ means dx=23sec2θ dθ, and turns gx into 2 secθ. From Figure 6.3.2, secθ=124+9 x2. Hence, the evaluation of the given integral proceeds as follows.
∫x 4+9 x2ⅆx
= ∫23tanθ2 secθ23sec2θ dθ
=4+9 x2274+9 x2
The integral in the second line is evaluated by writing it in the form ∫u2 dx by setting u=secθ. However, there is an "easier" solution to the original integral, obtained by the substitution u=4+9 x2 so that du=18 x dx. With this substitution, the integral becomes ∫u1/2 ⅆu18 whose value is
in agreement with the solution obtained by the trig substitution.
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫x 4+9 x2ⅆx = 127⁢9⁢x2+43/2
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≔∫x 4+9 x2ⅆx:
Change variables as per Table 6.3.1
Use the Change command to apply the change of variables x=23tanθ.
Simplify the radical to 2 secθ. Note the restriction imposed on θ.
(Maple believes that the sine and cosine functions are "simpler" so it resists writing the integrand as tanθsec3θ.)
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate the integral.
Revert the change of variables by applying the substitution θ=arctan3 x/2.
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 that results immediately in the integral 19∫u2 du.
On the other hand, Table 6.3.11(a) shows the result when the Change rule x=23tanθ is imposed on the tutor. The integrand is written as tanθsec3θ and its antiderivative is found by the Change rule with u=secθ.
Table 6.3.11(a) Annotated stepwise solution via Integration Methods tutor
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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