MATH100 Study Guide - Final Guide: The Sketch, Maxima And Minima, Riemann Sum

• School: University of Alberta
• Department: Mathematics
• Course: MATH100
• Professor: All

Document Summary

1. (a) introducing the substitution u = sin ((cid:18)), leads to du = cos ((cid:18)) d(cid:18) and that (cid:18) = 0 () u = 0 and (cid:18) = (cid:25) 4 (b) observing that the integrand is a continuous even function and that ln (1) = 0, implies ln(1)z (cid:0) ln(1) (cid:18)2 e(cid:18)2 cos(cid:16)e(cid:18)2(cid:17) d(cid:18) = 0z0 (cid:18)2 e(cid:18)2 cos(cid:16)e(cid:18)2(cid:17) d(cid:18) = 0: (c) we observe that. Z cos2 (x) sin (2x) dx = 2z cos3 (x) sin (x) dx: Introducing the substitution u = cos (x) =) du = (cid:0) sin (x) dx. =)z cos2 (x) sin (2x) dx = (cid:0)2z u3 du = (cid:0) u4. = (cid:0) (d) introducing the substitution cos4 (x) + c: u = 1 + x2 =) du = 2x dx; implying that. 2z (u (cid:0) 1)pu du = (cid:0)(cid:0)1 + x2(cid:1)3=2. 2 (cid:0) x =) du = (cid:0)dx, and x = 0 =) u = (cid:25)