Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x24+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.
= ∫23tanθ22 secθ23sec2θ dθ
=1627∫sec5θ dθ−∫sec3θ dθ
=162714sec3θtanθ+34∫sec3θ dθ−∫sec3θ dθ
=2274+9 x2232x2 4+9 x24−1−227ln4+9 x22+32x
=x364+9 x22+9 x2−227ln4+9 x22+32x
Line 3 is obtained by applying the trig identity tan2θ=sec2θ−1. Line 5 is obtained by applying the reduction formula derived in Example 6.2.9. The integral of sec3θ evaluated in line 7 is derived in Example 6.2.5. The absolute value in line 7 is dropped in line 8 because the argument of the logarithm is positive for the θ-interval defined in Table 6.3.1.
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫x24+9 x2ⅆx = 136⁢x⁢9⁢x2+43/2−118⁢x⁢9⁢x2+4−227⁢arcsinh⁡32⁢x
Using the appropriate identity in Table 2.10.4, and the obvious algebraic manipulations, the alternate form of the solution, namely,
x364+9 x22+9 x2−227ln4+9 x22+32x
can be obtained from the Maple solution.
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=23tanθ.
Simplify the radical to 2 secθ. Note the restriction imposed on θ.
(Maple believes that the sine and cosine functions are "simpler" than secants and tangents.)
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.17, below.
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 u=9 x2+4−3 x that results in the calculations shown in Table 6.3.14(a). Note that the final step of replacing u with 9 x2+4−3 x has been deleted because the resulting expression is not simplified by the tutor.
Table 6.3.14(a) Maple's stepwise approach to the given integral
On the other hand, Table 6.3.14(b) shows the result when the Change rule x=23tanθ is imposed on the tutor. The trig identity tan2θ=sec2θ−1 must then be imposed by the Rewrite rule. After a second and third invocation of the Rewrite rule, the integral is in the form shown in the Mathematical Solution, above. Maple's stepwise solution will now proceed to re-derive the reduction formula that was derived in Example 6.2.9, and to re-derive the integral of sec3θ that was derived in Example 6.2.5.
Table 6.3.14(b) Initial steps in an 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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