Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫4+9 x2xⅆx.
The substitution x=23tanθ means dx=23sec2θ dθ, and turns gx into 2 secθ. From Figure 6.3.2, secθ=124+9 x2 and cosθ=2/4+9 x2. Hence, the evaluation of the given integral proceeds as follows.
= ∫2 secθ23sec2θ dθ23tanθ
=2∫sec3θtanθ ⅆθ=2∫1cos3θsinθcosθ ⅆθ
=−2 arctanh24+9 x2+24+9 x22
=4+9 x2−2 arctanh24+9 x2
The identity 1u21−u2=11−u2+1u2 is established by the algebraic device of partial fraction decomposition that will be studied formally in Chapter 6.4. The antiderivative of 1/u2−1 is obtained from Table 3.10.1. The logarithmic form of the inverse hyperbolic tangent function is given in Table 2.10.4.
The substitution v2=4+9 x2 leads to an alternate solution not based on trig substitution. Since 2 v dv=18 x dx, the integral becomes
∫v⋅v dv9 xx
= ∫v2dv9 x2
= ∫1+1v−2−1v+2 ⅆv
= v+ln(|v−2|)− lnv+2
=4+9 x2+ln4+9 x2−2−ln4+9 x2+2
The conversion of v2v2−4 to 1+1v−2−1v+2 is again attributed to the algebraic technique of partial fraction decomposition.
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫4+9 x2xⅆx = 9⁢x2+4−2⁢arctanh⁡29⁢x2+4
Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,
4+9 x2+ln4+9 x2−2−ln4+9 x2+2
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 cosecants.)
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.13(a), 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 u2=9 x2+4 that results in the integral ∫u2u2−4 du=∫1+4u2−4 du=∫1+1u−2−1u−2 du, the last form being the result of a partial fraction decomposition.
On the other hand, Table 6.3.13(a) shows the result when the Change rule x=23tanθ is imposed on the tutor. The further application of the Change rule with u=cosθ gives a rational function that yields to a partial fraction decomposition.
Table 6.3.13(a) Initial steps in an annotated stepwise solution via Integration Methods tutor
Rather than simply evaluating each of the three integrals in the last line of Table 6.3.13(a), the tutor evaluates each by making detailed changes of variables. Clearly, the outcome has to be
−ln(u+1) +ln(u−1) +2/u
with u=cosθ=2/9 x2+4, as per Figure 6.3.2.
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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