MATH 2260 Worksheet
Han-Bom Moon
Worksheet Week 6
Section 8.1. ∼ 8.2.
• This worksheet is for improvement of your mathematical writing skill. Writing
using correct mathematical expression and steps is really important part of doing
math. Please print out this worksheet and try to solve problems, following given
steps.
• You don’t need to submit this worksheet. It is not homework.
• The steps in the solution represent important information you have to use in the
solution. It must be mentioned on your own solution.
1. Evaluate the integral
Z
xex dx.
(a) Find two functions f (x) and g(x) such that
• f (x)g 0 (x) = xex ,
• f 0 (x)g(x) is easier to compute the antiderivative.
f (x) = x
g 0 (x) = ex
f 0 (x) = 1
g(x) = ex
f (x)g 0 (x) = xex
f 0 (x)g(x) = ex
(b) Use the integration by parts formula
Z
Z
f (x)g 0 (x) dx = f (x)g(x) − f 0 (x)g(x) dx
to express the integral.
Z
xex dx = xex −
Z
ex dx
(c) Compute the integral.
x
xe −
Z
ex dx = xex − ex + C
1
MATH 2260 Worksheet
Han-Bom Moon
2. Evaluate the integral
Z
x sec2 x dx.
f (x) = x
g 0 (x) = sec2 x
f 0 (x) = 1
g(x) = tan x
f (x)g 0 (x) = x sec2 x
Z
f 0 (x)g(x) = tan x
Z
2
x sec x dx = x tan x −
Z
tan x dx = x tan x −
sin x
dx
cos x
= x tan x − (− ln | cos x|) + C = x tan x + ln | cos x| + C
Sometimes you need to use the integration by parts method several times.
3. Evaluate the integral
Z
x2 sin x dx.
g 0 (x) = sin x
f (x) = x2
f 0 (x) = 2x
g(x) = − cos x
f (x)g 0 (x) = x2 sin x
Z
f 0 (x)g(x) = −2x cos x
Z
2
x sin x dx = −2x cos x −
2
−2x cos x dx = −x cos x +
Z
2x cos x dx
Z
For
2x cos x dx,
f (x) = 2x
g 0 (x) = cos x
f 0 (x) = 2
g(x) = sin x
f (x)g 0 (x) = 2x cos x
Z
f 0 (x)g(x) = 2 sin x
Z
2x cos x dx = 2x sin x− 2 sin x dx = 2x sin x−(−2 cos x) = 2x sin x+2 cos x+C
2
−x cos x +
Z
2x cos xdx = −x2 cos x + 2x sin x + 2 cos x + C
2
MATH 2260 Worksheet
Han-Bom Moon
The next part of this worksheet contains a technique to compute products of sines
and cosines, which I will not mention in the class.
We start with several trigonometric identities.
sin(a + b) = sin a cos b + sin b cos a,
sin(a − b) = sin a cos b − sin b cos a
cos(a + b) = cos a cos b − sin a sin b,
cos(a − b) = cos a cos b + sin a sin b
By combining these two equalities, we can find several new identities:
(a) Show the following equality.
1
(cos(m − n)x − cos(m + n)x) = sin mx sin nx
2
1
(cos(m − n)x − cos(m + n)x)
2
=
1
(cos mx cos nx + sin mx sin nx − (cos mx cos nx − sin mx sin nx))
2
=
1
(2 sin mx sin nx) = sin mx sin nx
2
(b) There are two more equalities we can show in similar ways.
1
(sin(m − n)x + sin(m + n)x) = sin mx cos nx
2
1
(cos(m − n)x + cos(m + n)x) = cos mx cos nx
2
4. Evaluate
Z
sin 3x cos 5x dx
(a) Express the product as a sum of two trigonometric functions.
sin 3x cos 5x =
1
1
(sin(3 − 5)x + sin(3 + 5)x) = (sin(−2x) + sin 8x)
2
2
(b) Compute the integral.
Z
Z
1
(sin(−2x) + sin 8x) dx
sin 3x cos 5x dx =
2
1
=
2
1
1
1
1
(− cos(−2x)) + (− cos 8x) + C = cos(−2x) −
cos 8x + C
−2
8
4
16
3
MATH 2260 Worksheet
Han-Bom Moon
5. Evaluate
Z
cos 4x cos 2x dx
1
1
(cos(4 − 2)x + cos(4 + 2)x) = (cos 2x + cos 6x)
2
2
Z
Z
1
cos 4x cos 2x dx =
(cos 2x + cos 6x) dx
2
1
1
1
1 1
sin 2x + sin 6x + C = sin 2x +
sin 6x + C
=
2 2
6
4
12
cos 4x cos 2x =
6. Evaluate
Z
π
2
cos x cos 7x dx
− π2
cos x cos 7x =
Z
π
2
− π2
1
1
(cos(1 − 7)x + cos(1 + 7)x) = (cos(−6x) + cos 8x)
2
2
Z
cos x cos 7x dx =
π
2
− π2
1
(cos(−6x) + cos 8x) dx
2
π
π
2
2
1
1
1
1
1
=
sin(−6x) + sin(8x)
= − sin(−6x) +
sin(8x)
2 −6
8
12
16
− π2
− π2
1
1
1
1
= − sin(−3π) +
sin(4π) − − sin(3π) +
sin(−4π) = 0
12
16
12
16
4