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Math 1432
Section 15251
Aarti Jajoo
[email protected] (add Math 1432 in subject)
606 PGH
Office Hours in 606 PGH:
Starting from Wednesday
Monday 1:30 – 2:30 p.m.
Wednesday 1:30 – 2:30 p.m.
POPPER
1. x ln x 2 dx =
∫
a. x 2 ln x + C
x2
b. −
+C
2
x2
c. x ln x +
+C
2
2
x
d. x 2 ln x −
+C
2
e. None of these
2.
∫ arctan xdx
3.
∫
x 2 arctan xdx
4.
∫x
3 − x2
e
dx
Integration by parts with definite integrals.
∫
5.
∫
1
0
xe x dx
b
a
b
u dv = ( uv )| −
a
∫
b
a
v du
6.
π2
∫0
x 2 sin x dx
7.
∫( e
x
+ 2x
)
2
dx
Section 8.3
Powers and Products of Trigonometric Functions
Recall the following identities:
cos 2 ( x ) + sin 2 ( x ) = 1
1 + tan 2 ( x ) = sec 2 ( x )
1 + cot 2 ( x ) = csc 2 ( x )
cos
2
(x) =
1 + cos ( 2 x )
2
sin ( 2 x ) = 2 sin x cos x
cos ( 2 x ) = cos 2 x − sin 2 x
sin
2
(x) =
1 − cos ( 2 x )
2
In this section, we will study techniques for evaluating integrals of
the form
∫
m
n
sin x cos x dx
and
∫
m
n
sec x tan dx
where either m or n is a positive integer.
To find antiderivatives for these forms, try to break them into
combinations of trigonometric integrals to which you can apply the
Power Rule, which is
∫
⎧ u n+1
+C
⎪
n
u du = ⎨ n + 1
⎪ln u + C
⎩
if n ≠ 1
if n = − 1 .
Integrals Involving Powers of Sine and Cosine
1. If the power of the sine is odd and positive, save one sine
factor to be the du and convert the remaining factors to cosine.
Then, expand and integrate.
∫
sin 3 x dx
∫
sin 3 x cos 2 x dx
2. If the power of the cosine is odd and positive, save one
cosine factor to be the du and convert the remaining factors to
sine Then, expand and integrate.
∫
cos 5 x dx
∫
sin 4 x cos 5 x dx
3. If the powers of both the sine and cosine are even and
nonnegative, make repeated use of the half angle identities:
sin 2 x =
∫
cos 2 x dx
1 − cos 2 x
2
and
cos 2 x =
1 + cos 2x
.
2
∫
cos 4 x dx
4
2
cos
x
sin
x dx
∫
Integrals involving Secants and Tangents
1. If the power of the secant is even and positive, save a secant
squared factor to be the du and convert the remaining factors to
tangents. Then, expand and integrate.
∫
sec 4 x dx
∫
sec 4 x tan x dx
2. If the power of the tangent is odd and positive, and there is
a secant factor, save a secant-tangent factor to be the du and
convert the remaining factors to secants. Then, expand and
integrate.
∫
tan 3 x sec x dx
Compute
3
sec
2x
tan
(
)
( 2x ) dx
∫
If there is no secant factor, convert a tangent squared factor to a
secant squared factor; then expand and repeat if necessary.
∫
tan 3 x dx
3. If there are no secant factors and the power of the tangent
is even and positive, convert a tangent squared factor to a
secant squared factor; then expand and repeat if necessary.
∫
tan 4 x dx
m
sec
x dx where m is odd and
∫
positive, use integration by parts.
4. If the integral is of the form
∫
sec 3 x dx
5. If none of the first four cases applies, try converting to sines
and cosines.
sec x
∫ tan 2 x dx
Integrals involving Sine-Cosine Products
with Different Angles
Use the product-to-sum identities.
1
sin A sin B = cos ( A − B ) − cos ( A + B )
2
(
1
sin A cos B =
sin ( A − B ) + sin ( A + B )
2
(
)
1
cos A cos B = cos ( A − B ) + cos ( A + B )
2
(
)
)
Rewrite sin (5x) sin (3x)
∫ sin ( 5 x ) sin ( 3 x ) dx =