Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫14−9 x2 ⅆx.
The substitution x=23sinθ means dx=23cosθ dθ, and turns fx into 2 cosθ. Hence, the evaluation of the given integral proceeds as follows.
∫14−9 x2 ⅆx
= ∫23cosθ2 cosθ ⅆθ
= ∫13 ⅆθ
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫14−9 x2 ⅆx = 13⁢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≔∫14−9 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. Both the numerator and denominator become multiples of cos2θ.
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate what is now a very simple integral.
Revert the change of variables by applying the substitution θ=arcsin3 x/2.
Table 6.3.2(a) displays the annotated stepwise solution provided by the
Table 6.3.2(a) Annotated stepwise solution via Integration Methods tutor
The Change rule in the tutor subsumes all the simplifications by which the integrand becomes 1/3 after the change of variables has been made. The Change command in the IntegrationTools command leaves those simplifications for the user to perform.
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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