Download MAT137 | Calculus! | Lecture 30

MAT137 | Calculus! | Lecture 30
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Integration Methods:
§8.4 Trig. Substitution
§8.5 Rational Functions.
Integration Methods:
§8.3 Trig. Functions
official website http://uoft.me/MAT137
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Trigonometric Identities
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Derivatives of Trigonometric Functions
d
tan x = sec2 x
dx
d
sin x = cos x
dx
d
cos x = − sin x
dx
Beatriz Navarro-Lameda
d
sec x = sec x tan x
dx
L0601
MAT137
24 January 2017
Trigonometric Identities
Pythagorean Identities
sin2 x + cos2 x = 1
(1)
tan2 x + 1 = sec2 x
(2)
2
2
1 + cot x = csc x
(3)
Double-Angle Formulas
cos(2x) = cos2 x − sin2 x = 1 − 2 sin2 x
(4)
sin(2x) = 2 sin x cos x
(5)
Half-Angle Formulas
sin2 x =
1
2
− 21 cos(2x)
(6)
cos2 x =
1
2
+ 12 cos(2x)
(7)
You can obatain all these identities using a clever combination of (1) and (4).
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 1
Evaluate ∫ cos8 x sin x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 2
Evaluate ∫ cos3 x sin4 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 3
Evaluate ∫ sin5 x cos2 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 4
Evaluate ∫ sin3 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ sinn x cosm x dx:
if
if
, then try u = sin x;
, then try u = cos x.
If both m and n are odd, you can try either substitution.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ sinn x cosm x dx:
if m is odd, then try u = sin x;
if
, then try u = cos x.
If both m and n are odd, you can try either substitution.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ sinn x cosm x dx:
if m is odd, then try u = sin x;
if n is odd , then try u = cos x.
If both m and n are odd, you can try either substitution.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 5
Evaluate ∫ cos2 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 6
Evaluate ∫ sin4 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
We can use the same strategy to compute integrals involving powers of
tan x and sec x.
Homework: Try to evaluate the following integrals.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 7
Evaluate ∫ sec4 x tan5 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Example 8
Evaluate ∫ sec3 x tan3 x dx.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ secn x tanm x dx:
if m is odd, then try
Beatriz Navarro-Lameda
;
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ secn x tanm x dx:
if m is odd, then try
;
→ save a factor of sec x tan x and use tan2 x = sec2 x − 1 to express the
remaining factors in terms of sec x.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ secn x tanm x dx:
if m is odd, then try
;
→ save a factor of sec x tan x and use tan2 x = sec2 x − 1 to express the
remaining factors in terms of sec x.
if n is even, then try
Beatriz Navarro-Lameda
.
L0601
MAT137
24 January 2017
Integral of Trig. Functions
Method
To compute ∫ secn x tanm x dx:
if m is odd, then try
;
→ save a factor of sec x tan x and use tan2 x = sec2 x − 1 to express the
remaining factors in terms of sec x.
if n is even, then try
.
→ save a factor of sec2 x and use sec2 x = tan2 x + 1 to express the
remaining factors in terms of tan x.
Beatriz Navarro-Lameda
L0601
MAT137
24 January 2017