Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫14+9 x2 ⅆx.
The substitution x=23tanθ means dx=23sec2θ dθ, and turns gx into 2 secθ. From Figure 6.3.2, secθ=124+9 x2. Hence, the evaluation of the given integral proceeds as follows.
∫14+9 x2 ⅆx
= ∫23sec2θ dθ2 secθ
= 13∫secθ ⅆθ
In the final step, the absolute values in the logarithm are dropped because the argument of the logarithm is positive for all real x.
Evaluate the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
∫14+9 x2 ⅆx = 13⁢arcsinh⁡32⁢x
Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,
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.
Q≔∫14+9 x2 ⅆx:
Change variables as per Table 6.3.1
Use the Change command to apply the change of variables x=23tanθ.
Simplify the radical to 2 secθ. Note the restriction imposed on θ.
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate the integral.
Revert the change of variables by applying the substitution θ=arctan3 x/2.
Maple resists changing the integral of 1/cosθ to the integral of secθ.
The stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution u=9 x2+4−3 x, and ends with
an antiderivative that differs from the earlier solution by ln2, an additive constant of integration. Also, the argument of the logarithm has to be rationalized to make one expression look like the other.
On the other hand, Table 6.3.10(a) shows the result when the Change rule x=23tanθ, the sec rule, and the Revert rule are applied in the tutor.
Table 6.3.10(a) Annotated stepwise solution via Integration Methods tutor
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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