Chapter 6: Techniques of Integration
Section 6.6: Rationalizing Substitutions
Table 6.6.1 lists two common types of substitutions that turn an integrand into a rational function of a single variable, in which case the method of partial fractions would apply.
u=gxn or u=gx or un=gx
gxn and gxm
u=gxk, k = least common multiple of m and n
Rational function of sinx and cosx
z=tanx2,dx=2 dz1+z2,sinx=2 z1+z2,cosx=1−z21+z2
Table 6.6.1 Common rationalizing substitutions
The consequences of the substitution z=tanx/2 listed in Table 6.6.1 hinge on Figure 6.6.1 and the calculations to its right.
From z=tanx/2 and Figure 6.6.1 it follows that
sinx2=z1+ z2 and cosx2=11+ z2
From the trig identities
it follows that
sinx=2 z1+ z2 and cosx=1−z21+ z2
Since x=2 arctanz, then dx=2 dt1+z2
Figure 6.6.1 z=tanx2,z∈−π,π
Evaluate the indefinite integral ∫xx−1 ⅆx.
Evaluate the indefinite integral ∫1x1/3+x1/4 ⅆx.
Evaluate the indefinite integral ∫12 sinx+3 cosx ⅆx.
Evaluate the integral ∫12 sinx+3 cosx ⅆx without making the rationalizing substitution z=tanx/2.
Obtain a continuous antiderivative for 1/7+2 sinx+3 cosx.
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