Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫x 4−9 x2ⅆx.
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.
∫x 4−9 x2ⅆx
= ∫23sinθ⋅2 cosθ⋅23cosθ ⅆθ
= 89∫sinθcos2θ ⅆθ
= −827124−9 x23
= −1274−9 x23/2
The third line is obtained by setting u=cosθ so du=−sinθ dθ, and the integral becomes ∫u2 ⅆu.
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Simplify
∫x 4−9 x2ⅆx = 127⁢3⁢x−2⁢3⁢x+2⁢−9⁢x2+4= simplify −127⁢−9⁢x2+43/2
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≔∫x 4−9 x2ⅆx:
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
Use the value command to evaluate the integral, which is of the form u2 du, with u=cosθ.
Revert the change of variables by applying the substitution θ=arcsin3 x/2.
Table 6.3.3(a) displays the annotated stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules.
Table 6.3.3(a) Annotated stepwise solution via Integration Methods tutor
Note that the substitution chosen by Maple is not a trig substitution taken from Table 6.3.1.
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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