Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x24−9 x2 ⅆx.
The substitution x=23sinθ means dx=23cosθ dθ, and turns fx into 2 cosθ. From Figure 6.3.1, cosθ=124−9 x2. Hence, the evaluation of the given integral proceeds as follows.
∫x24−9 x2 ⅆx
= ∫23sinθ223cosθ dθ2 cosθ
= 427∫sin2θ ⅆθ
=427∫121−cos2 θ ⅆθ
=227 arcsin32x=x184−9 x2
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫x24−9 x2 ⅆx = −118⁢x⁢−9⁢x2+4+227⁢arcsin⁡32⁢x
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=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 sin2θ=1− cos2 θ/2 via the simplify command with the identity as a side relation.
q3≔simplifyq2,sin2θ=1− cos2 θ/2
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 simple integrals.
Apply the double-angle trig expansion to sin2 θ via the expand command.
Revert the change of variables by applying the substitution θ=arcsin3 x/2.
Table 6.3.7(a) displays the annotated stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules.
Table 6.3.7(a) Annotated stepwise solution via Integration Methods tutor
The essential difference between these two approaches is that Maple's stepwise rules will not permit the integration of cos2 u without first making the change of variables u1=2 u.
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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