264
Worksheet 7.1.
Integration by Parts
1. Use Integration by Parts to evaluate
Z
xe−x dx.
2
2. Use the substitution u = x and then Integration by Parts to evaluate
3. Compute the definite integral
Z
1
3
ln x dx.
Z
2
x3 ex dx.
265
π
4. Find the volume of the solid obtained by revolving y = cos x for 0 ≤ x ≤
about the
2
y-axis.
266
Solutions to Worksheet 7.2
1. Use integration by parts to evaluate
Z
xe−x dx.
Let u = x and v ′ = e−x . Then we have u = x, v = −e−x , u′ = 1, and v ′ = e−x . Using
Integration by Parts, we get
Z
Z
−x
−x
x e dx = x(−e ) − (1)(−e−x ) dx
Z
−x
= −xe + e−x dx = −xe−x − e−x + C
= −e−x (x + 1) + C
2
2. Use the substitution u = x then integration by parts to evaluate
Let w = x2 . Then dw = 2x dx, and
Z
Z
1
3 x2
wew dw.
x e dx =
2
Z
2
x3 ex dx.
Using Integration by Parts, we let u = w and v ′ = ew . Then we have
Z
Z
w
w
we dw = we − (1)ew dw = wew − ew
Substituting back in terms of x, we get
Z
1 2 x2
2
2
x3 ex dx =
x e − ex + C
2
3. Compute the definite integral
Z
3
ln x dx.
1
Let u = ln x and v ′ = 1. Then u′ = 1/x and v = x. Using Integration by Parts,
Z 3
ln x dx = x ln x − x|31 = 3 ln 3 − 3 − ((1) ln 1 − 1)
1
= 3 ln 3 − 2
π
about the
4. Find the volume of the solid obtained by revolving y = cos x for 0 ≤ x ≤
2
y-axis.
267
Use the Cylindrical Shells method, where the volume V is given by
Z b
Z π
2
V =
(2πr)h dx = 2π
x cos x dx
a
0
π
and the radius r = x and varies from 0 to , and the height h = y = cos x.
2
Using Integration by Parts, with u = x and v ′ = cos x, we get
i
h π
π
+ 0 − (0 + 1)
V = 2π [x sin x + cos x]02 = 2π
2
= π(π − 2)