Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x24+9 x2 ⅆx.
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.
∫x24+9 x2 ⅆx
= ∫23tanθ223sec2θ dθ2 secθ
=427∫sec3θ dθ−∫secθ dθ
=2274+9 x2232x−227ln4+9 x22+32x
=x184+9 x2−227ln4+9 x22+32x
Line 3 is obtained by applying the trig identity tan2θ=sec2θ−1. The integral of sec3θ evaluated in line 5 is derived in Example 6.2.5. The integral of secθ is derived in Table 6.2.10. The absolute values in line 5 are dropped in line 6 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 x2 ⅆx = 118⁢x⁢9⁢x2+4−227⁢arcsinh⁡32⁢x
Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,
x184+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.
Q≔∫x24+9 x2 ⅆx:
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 tangents and secants.)
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.15(b), below.
To revert the change of variables, apply the substitution θ=arctan3 x/2 via
Context Panel: Evaluate at a Point≻θ=arctan3⋅x/2
→evaluate at point
Control-drag the first two terms and press the Enter key.
Select these two terms (in the output) and select "normal" in the Smart Pup-Up at the top of the Context Panel.
Control-drag the result and to it, control-drag the log-term from the previous cell.
Press the Enter key.
From Figure 6.3.2, sinθ=3 x/9 x2+4, and cosθ=2/9 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 u=9 x2+4−3 x and proceeds as shown in Table 6.3.15(a).
Table 6.3.15(a) The substitution u=9 x2+4−3 x made by the Integration Methods tutor
Note how the tutor requires the Rewrite rule where at top-level the expand command would suffice. Note further that the solution in Table 6.3.18 is not complete - the Revert rule has not been applied. If it were, the result would be the following, a result in dire need of a simplification that cannot be effected in the tutor.
Table 6.3.15(b) shows the result when the Change rule x=23tanθ is imposed on the tutor.
Table 6.3.15(b) Initial steps in an annotated stepwise solution via Integration Methods tutor
Note the use of the Rewrite rule to effect transformations that at top-level could be implemented more directly. Note also that the two integrals in the last line of Table 6.3.15(b) are not evaluated. Maple's stepwise evaluation of these two integrals reproduces the derivations in Example 6.2.5 and Table 6.2.10, respectively.
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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