Lecture 3: Techniques for Integration
II. Trigonometric Integrals, part I: Solving integrals of the sine and cosine (7.2)
In this second integration technique, you will
study techniques for evaluating integrals of the
form
Z
Z
sinm x cosn x dx and
secm x tann x dx
Recall:
sin2 x + cos2 x = 1
cos2 x = 21 (1 + cos 2x) (Half angle formula)
sin2 x = 12 (1 − cos 2x) (Half angle formula)
sin(2x) = 2 sin x cos x (Sine double angle)
1
EVEN Power of sin x−
Evaluate
Z
ex.
sin2 xdx
Z
ex.
( 12 (x − 12 sin(2x)) + c)
sin4 xdx
1
( 83 x − 14 sin(2x) + 32
sin(4x) + c)
Similar technique applies to EVEN power of
the cos x.
2
ODD power of sin x−
Evaluate
Z
ex.
sin3 x dx
(− cos x +
cos3 x
3
+ c)
Similar technique applies to ODD power of cos x.
3
ODD power of sin x or cos x in the product:
Evaluate
Z
ex.
sin5 x cos2 xdx
7
(− cos7
(ODD before EVEN)
4
x
+
2
5
cos5 x −
1
3
cos3 x + c)
EVEN power of sin x and cos x in the product:
Evalluate
Z
ex.
sin4 x cos2 xdx
1
( 16
x−
1
64
sin(4x) −
1
48
sin3 (2x) + c)
5
Z
sinm x cosn xdx
(Odd before Even)
1. When the power of the sine is odd and positive, save one sine factor and convert the remaining even power factor to cosine.
Then use u−sub and integrate.
Z z }| {
Z
z }| {
z }| {
2k
+
1
n
2
k
n
x cos x dx = (sin x) cos x sin x dx
sin
R
= (1−cos2 x)k cosn x sin x dx
2. When the power of the cosine is odd and positive, save one cosine factor and convert the
remaining even power factor to sine.
Then use u−sub and integrate.
Z
Z
z }| {
z }| { z }| {
sinm x cos2k + 1 x dx = sinm x (cos2 x)k cos x dx
=
6
R
sinm x(1−sin2 x)k cos x dx
3. When the powers of both the sine and cosine
are even and nonnegative, make use of the
half-angle formula and/or the Sine double angle
formula.
Z
NYTI: Evaluate
sin5 x cos3 xdx
6
8
( sin6 x − sin8 x +c)
NOTE: In this class, we will learn how to integrate sin x and cos x problems without using the
reduction formulas in section 7.2.
7
Integrate product of the Sine and Cosine with different angles
Recall
sin A cos B = 21 [sin(A − B) + sin(A + B)]
sin A sin B = 21 [cos(A − B) − cos(A + B)]
cos A cos B = 21 [cos(A − B) + cos(A + B)]
R
sin(4x) cos(5x)dx ( cos(x) − cos(9x) + c)
ex.
1
2
8
1
18