Chapter 6: Techniques of Integration
Section 6.3: Trig Substitution
Integrals containing one of the expressions on the left in Table 6.3.1 may yield to the companion substitution suggested in the middle column of the table. The substitution, called a trig substitution, is based on the related trig identity stated in the rightmost column of the table.
x=ab secθ,θ∈0,π2 or θ∈π,32π
Table 6.3.1 Trig substitutions
Figures 6.3.1 - 6.3.3 are useful for expressing the basic trig functions in terms of the variables and parameters appearing in the substitutions listed in Table 6.3.1.
Figure 6.3.1 For x=ab sinθ
Figure 6.3.2 For x=ab tanθ
Figure 6.3.3 For x=ab secθ
Table 6.3.2 lists eight examples for each case given in Table 6.3.1. Each of the three cases in Table 6.3.1 is represented by one of the functions defined at the top of Table 6.3.2. For each of these representative functions, the examples chosen are common instances of indefinite integrals in which the given functions appear. Notice the pattern of the eight examples in each column of Table 6.3.2: The function and its reciprocal, the function multiplied by x, divided by x, and divided into x, then multiplied by x2, divided by x2, and divided into x2.
∫x fx dx
∫x gx dx
∫x hx dx
∫x2 fx dx
∫x2 gx dx
∫x2 hx dx
Table 6.3.2 Examples in which the trig substitutions of Table 6.3.1 are to be used
Note that while gx is real for all real x, fx is real for x≤2/3, and hx is real for x≥2/3.
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