Download Integrals Potpourri

Integrals
100
Answer
Question
200
Answer
Question
300
Answer
Question
OR
400
Answer
Question
500
Answer
Question
The solution to
Z
√
x
dx.
4 − x2
√
What is − 4 − x2 + C?
Z
The solution to (ln x)2 dx.
[[usub with u = 4 − x2 ]]
What is x(ln x)2 − 2(x ln x − x) + C?
The solution to
1
10
R
[[parts with u = (ln x)2 , dv = dx]]
tan3 x sec8 x dx.
sec10 x −
1
8
R
[[ (sec2 x − 1) sec7 x(sec x tan x) dx]]
R
What is tan x + tan3 x + 35 tan5 x + 17 tan7 x + C? [[ tan3 x(tan2 x + 1)3 sec2 x dx]]
Z
2x3 − x2 + 2
The solution to
dx.
x3 + 1
R
2
What is 2x − 13 ln |x3 + 1| + C?
[[ (2 − x3x+1 ) dx]]
Z
3x3 + 2x + 3
The solution to
dx.
x2 (x2 + 1)
R
What is 2 ln |x| − 3x−1 + 21 ln |x2 + 1| − 3 tan−1 x + C?
[[ ( x2 + x32 + xx−3
2 +1 ) dx]]
What is
sec8 x + C?
Potpourri
100
Answer
Question
200
Answer
Question
300
Answer
Question
400
Answer
Z
An integral that becomes
u2 du after a u-substitution.
Z
What is
sin2 x cosx dx?
[[many possible questions]]
Z
x
An integral that becomes x tan−1 x −
dx after integration by parts.
1 + x2
Z
What is
tan−1 x dx?
[[u = tan−1 x, dv = dx]]
The algebraic function that is equal to tan2 (sin−1 x2 ).
What is
x2
?
4 − x2
x
)2 ]]
[[( √4−x
2
The form of the partial fraction decomposition for
(do not solve for A, B, etc.)
Question
500
Answer
What is
Ax + B
Cx + D
E
F
+
+
+
?
x2 + x + 1 (x2 + x + 1)2
x−1 x+2
The sigma notation formula for the Trapezoid Rule for approximating
What is
n
X
f (xk−1 ) + f (xk )
k=1
[[factor!]]
Z
a
with n trapezoids.
Question
x4 + 1
.
(x2 + x + 1)2 (x2 + x − 2)
2
∆x?
b
f (x) dx
How-To
100
Answer
Question
200
Answer
Question
300
Answer
Question
400
Answer
Question
500
Answer
Question
The u-subsitution for integrating
Z
x
dx.
1 + x4
What is u = x2 ?
R
[[
The parts u and dv for integrating
What are u = ln x and dv =
1
x2
Z
1
1+u2
du]]
ln x
dx.
x2
dx?
√
cos3 x sin x dx.
The u-substitution (after rewriting) for integrating
Z
What is u = sin x?
√
R
[[ (1 − sin2 x) sin x cos x dx]]
The parts u and dv for integrating
Z
x3 sec2 x2 dx.
What are u = x2 and dv = x sec2 x2 dx?
The trigonometric substitution you would use for
What is x − 2 =
√
Z
5 sin u?
√
1
dx.
1 + 4x − x2
R
[[ √
1
]]
5−(x−2)2
Diffy Q
100
Answer
Question
200
Answer
Question
300
Answer
Question
400
Answer
Question
OR
500
Answer
Question
The solution to the initial value problem
dy
dx
= 4y, y(0) = 2.
What is y = 2e4x ?
The integrating factor for the differential equation
1
2
[[y 0 + x3 y = 13 x(x2 + 1)]]
What is e 6 x ?
The separated form of the differential equation
What is
xy + 3y 0
= x.
x2 + 1
dy
dx
+ x = 2xy.
1
dy = x dx?
2y − 1
dy
[[ dx
= x(2y − 1)]]
The family of solutions to the differential equation y 0 + 2xy = x.
What is y =
1
2
2
+ Ce−x ?
2
2
2
What is y = e−x ( 21 ex + C)?
The solution to the initial value problem
What is y =
−1
?
ln |x| − 12
2
2
[[y 0 ex + 2xex y = xex ]]
dy
dx
=
y2
x ,
y(1) = 2.
[[ y12 dy =
1
x
dx]]
Numbers
100
Answer
The
Z 3 decimal number that is equal to the Right Hand Sum approximation of
x2 dx with n = 4 (calculator permitted).
1
Question
200
Answer
[[((1.5)2 + 22 + (2.5)2 + 32 )( 21 )]]
What is 10.75?
The (most simplified) number that is equal to
Z
1
x2 ex dx.
0
Question
300
Answer
What is e − 2?
[[parts twice]]
Z 6
√
The decimal number that is equal to the Midpoint Sum approximation of
x dx
0
with n = 3 (calculator permitted).
Question
400
Answer
Question
500
Answer
√
√
√
[[( 1 + 3 + 5)(2)]]
What is 9.79796?
The (most simplified) number that is equal to
Z
2
√
√
2
4 − x2
dx.
x2
What is 1 − π4 ?
[[trig sub x = 2 sin u]]
The smallest integer n so that an approximation of
Z
5
x3 dx with n trapezoids will
2
be guaranteed accurate to within 0.01 (calculator permitted).
Question
3
[[ (5−2)
12n2 (30) < 0.01]]
What is n = 83?
Final Jeopardy: Integration
???
Answer
Question
The solution to
Z
What is 38 x +
sin(2x) +
1
4
cos4 x dx.
1
32
sin(4x) + C?
R
[[ ( 1+cos(2x)
)2 dx]]
2