Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫4−9 x2xⅆx.
Note that there is a discontinuity in the integrand at x=0, which is in −23,23, the interval for which the integrand remains real.
The substitution x=23sinθ means dx=23cosθ dθ, and turns fx into 2 cosθ. From Figure 6.3.1, cosθ=124−9 x2, cotθ=13 x4−9 x2, and cscθ=23 x. Hence, the evaluation of the given integral proceeds as follows.
= ∫2 cosθ⋅23cosθ dθ23sinθ
= 2∫cos2θsinθ ⅆθ
= −2 lncscθ+ cotθ+2 cosθ)
= −2 ln(23 x+4−9 x23 x)+4−9 x2
The antiderivative of cscθ is obtained in Table 6.2.10.
Evaluate the given integral
Control-drag the integral.
Evaluate and Display Inline
∫4−9 x2xⅆx = −9⁢x2+4−2⁢arctanh⁡2−9⁢x2+4
Recall that the given integrand is defined over the reals only when x∈0,2/3⋃−2/3,0. The solution given by Maple is not real for this domain because on this domain 2/4−9 x2≥1. (The inverse hyperbolic tangent function is complex for such arguments.)
Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,
−2 ln(23 x+4−9 x23 x)+4−9 x2
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=23sinθ.
Simplify the radical to 2 cosθ. Note the restriction imposed on θ.
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Apply the half-angle trig identity cos2θ=1+cos2 θ/2 via the simplify command with the identity as a side relation.
Split the integral with the expand command, while preventing Maple from applying the double-angle expansion to cos2 θ.
Use the value command to evaluate what are now two separate but known integrals. (See Table 6.2.10 for ∫dθsinθ=∫cscθ ⅆθ.)
Revert the change of variables by applying the substitution θ=arcsin3 x/2.
Without a restriction on θ, the logarithm in the penultimate step could be complex. Its argument is nonnegative only for θ∈0,π/2. It would be better to apply the logarithm to the absolute value of the arguments in this and the last step.
Table 6.3.5(a) displays the annotated stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules.
Table 6.3.5(a) First steps of an annotated stepwise solution via Integration Methods tutor
Maple's stepwise solution will next re-derive the antiderivative of cscu as per Table 6.2.10.
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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