Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Evaluate the indefinite integral ∫4+9 x2x2ⅆx.
The substitution x=23tanθ means dx=23sec2θ dθ, θ=arctan32x, and turns gx into 2 secθ. Hence, the evaluation of the given integral proceeds as follows.
= ∫2 secθ23sec2θ dθ23tanθ2
=3∫sec3θtan2θ dθ =3∫1cosθ3sinθcosθ2 dθ
=32ln(u+1) −32ln(u−1) −3u
=32lnsinθ+1 −32lnsinθ−1 −3sinθ
=32ln3 x4+9 x2+1 −32ln(3 x4+9 x2−1)−4+9 x2x
=32ln(3 x+4+9 x23 x−4+9 x2)−4+9 x2x
=32ln(3 x+4+9 x22−4)−4+9 x2x
=3 ln123 x+4+9 x2−4+9 x2x
The identity 1u21−u2=1/2u+1−1/2u−1+1u2 is established by the algebraic device of partial fraction decomposition, that will be studied formally in Section 6.4. The absolute value in the logarithm of sinθ−1 can't be dropped since this quantity can be negative, whereas sinθ+1 is nonnegative.
Figure 6.3.2 can be used to obtain sinθ=3 x/4+9 x2.
The last three lines of the solution given above are the result of algebraic manipulations. First, the properties of real logarithms are used to combine the two log terms to one. Then, rationalization of the denominator in the argument of this single log term is used to simplify the argument. The result is seen in the penultimate line. The simplification in the final line is again the result of applying a property of real logarithms.
Evaluate the given integral
Control-drag the integral and press the Enter key.
Select the first two terms in the antiderivative and choose "normal" in the smart pop-up.
Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,
3 ln123 x+4+9 x2−4+9 x2x
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.
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.
(Maple believes that the sine and cosine functions are "simpler" than secants and cosecants.)
q2≔simplifyq1 assuming θ∷RealRange−π2,π2
Use the value command to evaluate the integral, or follow the approach in Table 6.3.16(b), below.
To revert the change of variables, apply the substitution θ=arctan3 x/2 via
Context Panel: Evaluate at a Point≻θ=arctan3⋅x/2
→evaluate at point
From Figure 6.3.2, sinθ=3 x/9 x2+4, and secθ=129 x2+4.
The stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution u2=9+4/x2 and proceeds as shown in Table 6.3.16(a).
Table 6.3.16(a) The substitution u2=9+4/x2 made by the Integration Methods tutor
Note that the solution in Table 6.3.16(a) is not complete - the antiderivatives have not been obtained and the Revert rule has not been applied. Indeed, the final result, a result in dire need of a simplification that cannot be effected in the tutor, is
Table 6.3.16(b) shows the result when the Change rule x=23tanθ is imposed on the tutor.
Table 6.3.16(b) Solution via Integration Methods tutor after x=23tanθ is imposed
The substitution u=sinθ leads to a rational function in u, an expression that is resolved by the algebraic technique of the partial fraction decomposition, to be studied in detail in Section 6.4.
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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