Substitution Sampler 1. Below, each entry in the left-hand column includes two blank spaces, seperated by a comma. Use letters and roman numerals from the middle and right-hand and columns to fill in the blanks. To do this, match the integrals in the left-hand column with useful identities and substitutions from the middle and right-hand columns. Choosing from the right-hand column might get easier if you choose from the middle column first. Each identity or substitution is used just once so, process of elimation might help. , a. (2 pts) Z 2 3 sin (x) cos (x) dx b. (2 pts) Z , 4 Z 1 + cot2 (x) = csc2 (x) B. Any of these... 2 sin(mx) sin(nx) = sin(mx) cos(nx) = cos(mx) cos(nx) = cos((m+n)x)−cos((m−n)x) 2 sin((m+n)x)+sin((m−n)x) 2 cos((m+n)x)+cos((m−n)x) 2 , II. x = sec(u) 3 III. sin(2x) cos(x) dx d. (2 pts) I. sec2 (x) − 1 = tan2 (x) sin (x) cos (x) dx c. (2 pts) Z A. sin2 (x) = 2 = cos (x) , cot4 (x) csc4 (x) dx du = 3dx C. Any of these... 1 2 (1 1 2 (1 − cos(2x)) + cos(2x)) IV. du = 2dx D. du = − csc2 (x)dx e. (2 pts) Z p x2 , − 9 dx V. E. du = cos(x)dx cos2 (u) = 1 − sin2 (u)