Substitution Sampler – Answers
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
E, V
2
sin (x) cos (x) dx
b. (2 pts)
Z
C , IV
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
B, III
sin(2x) cos(x) dx
d. (2 pts)
I.
sec2 (x) − 1 = tan2 (x)
sin (x) cos (x) dx
c. (2 pts)
Z
3
A.
III.
du = 3dx
C. Any of these...
sin2 (x)
=
2
=
cos (x)
D, I
cot4 (x) csc4 (x) dx
II.
x
= sec(u)
3
1
2 (1
1
2 (1
− cos(2x))
+ cos(2x))
IV.
du = 2dx
D.
du = − csc2 (x)dx
e. (2 pts)
Z p
x2
A, II
− 9 dx
V.
E.
du = cos(x)dx
cos2 (u) = 1 − sin2 (u)