Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫4−9 x2x2ⅆx.
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, and cotθ=4−9 x23 x. Hence, the evaluation of the given integral proceeds as follows.
= ∫2 cosθ23cosθ dθ23sinθ2
= −34−9 x23 x+arcsin32x
Evaluate the given integral
Control-drag the integral and press the Enter key.
Context Panel: Simplify≻Simplify
Context Panel: Expand≻Expand
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
Impose the trig identity cos2θ=1−sin2θ.
Use the expand command to split the fraction in the integrand into two separate fractions.
Use the trigsubs command to rewrite 1/sin2θ as csc2θ.
Use the expand command to split the integral into two separate integrals.
Use the value command to evaluate what are now two separate but relatively simple integrals.
Revert the change of variables by applying the substitution θ=arcsin3 x/2.
Table 6.3.8(a) displays the annotated stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules.
Table 6.3.8(a) First steps of an annotated stepwise solution via Integration Methods tutor
After the change of variables suggested by Table 6.3.1, Maple returns the integrand as cot2θ instead of as cos2θ/sin2θ. Obviously, this saves several manipulative steps. It then takes an application of the Rewrite rule to impose the trig identity cot2θ=csc2θ−1, at which point the calculation proceeds as expected.
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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