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Ch 7.2 Wkst AP Calculus BC
Name:
Integration by Parts
Objective: to integrate when u-substitution does not work
Ex 1:

x 3e x dx
Ex 2: Use the Tabular Method to evaluate

x 3 sin x dx
Ex 3:

x 2 ln x dx
Ch 7.2 Wkst AP Calculus BC
Name:
Integration by Parts
For problems #1 to #7, find the integrals by Integration by Parts.
1.
x
2.

2 x
e dx
x 3 cos 2 x dx
3.
x
4.

4
ln x dx
ln x
x10
dx
5.
 arctan x dx
6.
e
x
cos x dx
7.
 x arcsin x
2
dx
Hint: Do a normal u-substitution first,
then integrate by parts.
8.
 sec
3
x dx
Hint: Write as sec x sec ² x first, then integrate by parts. You’ll also need
to do a trigonometric substitution using the identity 1 + sec ² x = tan ² x.
9.
Given the initial condition y(0)  4 , find the particular solution to the differential equation
dy
x

sin x .
dx 2 y
10.
dy
1
) , then when x = 2 the solution
 (3x 2  ln x) y 2 and its solution curve passes through ( 1 ,
dx
2
curve passes through y =
If
(A)
1
4  ln 2
(B)
1
4  2 ln 2
(C) 
1
4  ln 2
(D) 
1
4  2 ln 2
(E) 
1
6  2 ln 2
11.
dy
 x e x and f (0)  2 . Find all values of x in the interval [ 0 , 3 ] for which the particular
dx
solution y  f (x) equals its average value.
CALCULATOR REQUIRED.
Suppose
(A) 1.501
(B) 1.624
(C) 1.998
(D) 2.615
(E) 2.332
ANSWERS:
1)
x2 e x  2x e x  2 e x  C
2)
1 x 3 sin 2 x  3 x 2 cos 2 x  3 x sin 2 x  3 cos 2 x  C
2
4
4
8
1 x 5 ln x  1 x 5  C
5
25
3)
4)
 1 x 9 ln x  1 x 9  C
9
81
5)
x arctan x  1 ln 1  x 2  C
2
6)
1 e x sin x  1 e x cos x  C
2
2
7)
1 x 2 arcsin x 2  1 1  x 4  C
2
2
1 sec x tan x  1 ln sec x  tan x  C
2
2
8)
9)
10) D
11) C
y   x cos x  sin x  16