Math%104%–Calculus % …shilinyu/teaching/Math104Sp...3. sin sin∫ (mx nx dx) ( ) ∫cos cos(mx...
Transcript of Math%104%–Calculus % …shilinyu/teaching/Math104Sp...3. sin sin∫ (mx nx dx) ( ) ∫cos cos(mx...
Math 104 -‐ Yu
Math 104 – Calculus 8.3 Trigonometric Integrals
Math 104 -‐ Yu
Three types of integrals
Nicolas Fraiman Math 104
Three types of integrals1.
!
2.
!
3.
Zsin
mx · cosn x dx
Ztanm x · secn x dx
Zsin(mx) sin(nx) dx
Zcos(mx) cos(nx) dx
Zsin(mx) cos(nx) dx
Math 104 -‐ Yu
Products of Sine and Cosine (m,n are posiEve integers) a) m, n: one or both = 1 Use u-‐subsEtuEon. Let u = the trig funcEon with power ≠ 1. b) m, n: one (or both) odd (both greater than 1)
1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into
the other trig funcEon. 3. Use u-‐subsEtuEon to finish the problem (let u = the “other” trig
funcEon) c) m, n: both even Replace all even powers using the double-‐angle formula: Nicolas Fraiman
Math 104
Products of sine and cosineIntegrals of the form:
A) m or n odd 1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into the other function. 3. Use substitution to finish the problem.
B) m and n even Replace all powers using the double-angle formulas
Zsin
mx · cosn x dx
sin
2x+ cos
2x = 1
sin
2x =
12 (1� cos 2x) and cos
2x =
12 (1 + cos 2x)
Nicolas Fraiman Math 104
Products of sine and cosineIntegrals of the form:
A) m or n odd 1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into the other function. 3. Use substitution to finish the problem.
B) m and n even Replace all powers using the double-angle formulas
Zsin
mx · cosn x dx
sin
2x+ cos
2x = 1
sin
2x =
12 (1� cos 2x) and cos
2x =
12 (1 + cos 2x)
Nicolas Fraiman Math 104
Products of sine and cosineIntegrals of the form:
A) m or n odd 1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into the other function. 3. Use substitution to finish the problem.
B) m and n even Replace all powers using the double-angle formulas
Zsin
mx · cosn x dx
sin
2x+ cos
2x = 1
sin
2x =
12 (1� cos 2x) and cos
2x =
12 (1 + cos 2x)
Math 104 -‐ Yu
Example 1.
Nicolas Fraiman Math 104
Examples1. Evaluate 2. Evaluate 3. Evaluate
!
4. Evaluate
Zcos
5x sin
2x dx
Z ⇡
0cos
4x sin
2x dx
Z ⇡
0cos
10x sinx dx
Zsin
7x cos
3x dx
Math 104 -‐ Yu
Example
Nicolas Fraiman Math 104
Examples1. Evaluate 2. Evaluate 3. Evaluate
!
4. Evaluate
Zcos
5x sin
2x dx
Z ⇡
0cos
4x sin
2x dx
Z ⇡
0cos
10x sinx dx
Zsin
7x cos
3x dx
Math 104 -‐ Yu
Example 3.
Nicolas Fraiman Math 104
Examples1. Evaluate 2. Evaluate 3. Evaluate
!
4. Evaluate
Zcos
5x sin
2x dx
Z ⇡
0cos
4x sin
2x dx
Z ⇡
0cos
10x sinx dx
Zsin
7x cos
3x dx
Math 104 -‐ Yu
Products of tangent and secant
c) Other cases: there is no set method...
b) n even
1. Factor out (sec
2x) which is the derivative of tanx.
2. Use sec
2x = tan
2x+1 to transform the remaining even power of secx
into tanx
3. Use u-substitution to finish the problem (u = tanx)
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Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions ( )tan sec , positive integersm nx x dx m n∫
) ( )B the power of sec :n x even 2 4: tan secex x xdx∫21. Factor out sec x
2 4 2 2 2: tan sec t sec sn ca eex x x xx x=
2 22. If 2, use sec 1 tan to transform the remaining even power
of sec to be in terms of tan
n x x
x x
> = +
( )2 2 2 2 2 2: tan sec sec sec1t n tanaex x x x x xx= +
( )3. use substitution to finish the problem let tanu u x! =
( )22 24 2: tan sec tan 1 tan s ecxex x xdx x dxx+=∫ ∫2
tan
sec
u x
du xdx
=
=( )2 21u u du= +∫ ( )2 4u u du= +∫
3 5
3 5u u C= + +
3 51 13 5
tan tanx x C= + +
Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions tan sec m nx x dx∫
)C For all other cases, there is no set method Here are some examples:
: tanex xdx∫sin
cos
xdx
x= ∫
1 du
u
!∫
cos
sin
u x
du xdx
=
= !
ln u C= ! +
ln cos x C= ! +
1ln cos x C
!= +
tan ln secxdx x C= +∫
: secex xdx∫convenient version of 1
sec tansec
sec tan
x xx dx
x x
+= "
+∫2sec sec tan
sec tan
x x xdx
x x
+=
+∫
( )2
sec tan
sec tan sec
u x x
du x x x dx
= +
= +sec ln sec tanxdx x x C= + +∫1
duu∫ ln u C= +
Rtanm x · secn xdx
a) m odd, n positive
1. Factor out (tanx secx), which is the derivative of secx.
2. Use tan
2x = sec
2x � 1 to transform the remaining even power of
tanx into secx.
3. Use u-substitution to finish the problem (u = secx)
Math 104 -‐ Yu
Example 4.
Nicolas Fraiman Math 104
Examples5. Evaluate 6. Evaluate
Ztan3 x sec3 x dx
Ztan2 x sec4 x dx
Math 104 -‐ Yu
Example 5.
Nicolas Fraiman Math 104
Examples5. Evaluate 6. Evaluate
Ztan3 x sec3 x dx
Ztan2 x sec4 x dx
Math 104 -‐ Yu
Example Other cases: 6. Evaluate
7. Evaluate
Nicolas Fraiman Math 104
ExamplesFor the remaining cases (m even, n odd) there is no method.
7. Evaluate and
Ztanx dx
Zsecx dx.
Nicolas Fraiman Math 104
ExamplesFor the remaining cases (m even, n odd) there is no method.
7. Evaluate and
Ztanx dx
Zsecx dx.
Math 104 -‐ Yu
Linear factors of sine and cosine a) If m≠n, change the product into a sum using the following
idenEEes:
b) If m=n, the formulas above become double-‐angle formulas
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Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions
( ) ( )3. sin sinmx nx dx∫ ( ) ( )cos cosmx nx dx∫ ( ) ( )sin cosmx nx dx∫, rational with m n m n!
We change the product into a sum using the following identities:
( ) ( ) [ ]( ) [ ]( )1
sin sin cos cos2
mx nx m n x m n x = " + "
( ) ( ) [ ]( ) [ ]( )1
cos cos cos cos2
mx nx m n x m n x = " + +
( ) ( ) [ ]( ) [ ]( )1
sin cos sin sin2
mx nx m n x m n x = " + +
( ) ( )sin 3 cos 5x x dx∫ [ ]( ) [ ]( )1
sin 3 5 sin 3 52
x x dx = " + ∫ +
( ) ( )1
sin 2 sin 82
x x dx= " + ∫ ( ) ( )1
sin 8 sin 22
x x dx= " ∫( ) ( )1 1
8 2
1cos 8 cos 2
2x x C= " + + ( ) ( )1 1
16 4cos 8 cos 2x x C= " + +
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Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions
( ) ( )3. sin sinmx nx dx∫ ( ) ( )cos cosmx nx dx∫ ( ) ( )sin cosmx nx dx∫, rational with m n m n!
We change the product into a sum using the following identities:
( ) ( ) [ ]( ) [ ]( )1
sin sin cos cos2
mx nx m n x m n x = " + "
( ) ( ) [ ]( ) [ ]( )1
cos cos cos cos2
mx nx m n x m n x = " + +
( ) ( ) [ ]( ) [ ]( )1
sin cos sin sin2
mx nx m n x m n x = " + +
( ) ( )sin 3 cos 5x x dx∫ [ ]( ) [ ]( )1
sin 3 5 sin 3 52
x x dx = " + ∫ +
( ) ( )1
sin 2 sin 82
x x dx= " + ∫ ( ) ( )1
sin 8 sin 22
x x dx= " ∫( ) ( )1 1
8 2
1cos 8 cos 2
2x x C= " + + ( ) ( )1 1
16 4cos 8 cos 2x x C= " + +
Math 104 -‐ Yu
Example 8.
Nicolas Fraiman Math 104
Examples!
8. Evaluate Z
sin(3x) cos(5x) dx