Math 1220
Midterm 1
October 1, 2015
Please answer all the questions below. The value of every question is
indicated at the beginning of it. You may only use scratch paper and a
non-graphing calculator. No cell phones, notes, books or music players are
allowed during the exam.
Name:
UID:
1. (10 pts) Compute the integral
Z
p
cos x 4 − sin2 x dx
You don’t need to undo your changes of variable at the end.
2. (10 pts) Compute the integral
Z
x
√
dx
x+5
3. (10 pts) Compute the integral
Z
√
x+1
dx
4 − 9x2
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4. (10 pts) Compute the integral
Z
x3 + x
dx
x+1
5. (10 pts) Solve the following linear differential equation
dy
+ 2y = x
dx
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6. (10 pts) Compute the integral
Z
√
cos3 x sin x dx
7. (5 pts) Write down a partial fraction decomposition for the following function
f (x) =
6x2 − 15x + 22
x3 (x + 3)(x2 + 1)(x2 + x + 1)2
Don’t try to find the constants: simply write down the pattern.
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8. (15 pts) Show that
Z
π/2
0
and use it to compute
R π/2
0
n−1
sin x dx =
n
n
sin7 x dx
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Z
0
π/2
sinn−2 x dx
9. (10 pts) Consider the function
Z
x
sin
f (x) =
0
πt2
2
dt,
0≤x≤2
1. Over which intervals is it increasing / decreasing?
2. Over which intervals is it concave up / down?
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10. (Extra credit) Suppose
that f is continuousRand strictly increasing on [0, 1] with f (0) = 0
R1
1
and f (1) = 1. If 0 f (x) dx = 25 , calculate 0 f −1 (y) dy.
Hint: There are two approaches to this: youR can either draw a picture or you can use
1
the change of variable y = f (x) to compute 0 f −1 (y) dy.
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