Evaluate the following definite integral using the method of Example 10.2 (page 67): 90 0 x +1 dx for a real with a > 0. Hint: Since the integrand is an even function of x you may use * cos(ax) dx cos (ax) - Pj x +1 -dx x +1 where P denotes the Cauchy principal value (as in lecture notes, Eq. (10.4)). For a>0 you may close the integration contour -RR in the upper complex half-plane. To show that the contour integral over the half-circle CR with radius R vanishes for R, you may use again 2 +12R-1 for ze C, and in addition |ea|=e" ay1 for z = C, where y = Im (z). Ex. 10.2. Compute xsinx x+2x+2 Let dx. z Z f(2)= 22 +22+2 (z-21)(z- zj) where 2-1+1. Note: z is a simple pole of f(z)elz in the upper-half plane with residue b - Res f(z)elz= 21 elz 21-2 Use the contour C= Lg + CR shown in Fig. 10.2. We see xelx dx=2rib-f(z)e dz +2x+2 R want to bound this Im z CR R Figure 10.2: Contour for Ex. 10.2 Rez (10.16) (10.17) (10.18) (10.19) Note: If(z)| Mg where MR = R/(R-2)2 and le=ey