Calculus Lecture - Ch 8
8.1
Overview of
Integration Methods
8.2 Integration by Parts
Rule
Derivation: differentiate u*v with respect to x
Repeated application: problem with nested parentheses
Tabular method
Cyclers
ln, and inverse trig functions: integrate 1 dx, recalculate after one step
Worksheet
8.3 Trigonometric Integrals
tan(x) or cot(x) alone:
integral of tan(x) is ln|sec(x)| or -ln|cos(x)| Memorize or derive every time.
integral of cot(x) is ln|sin(x)|
sec(x) or csc(x) alone: integrating factor: ln|sec(x)+tan(x)| or ln|csc(x)-cot(x)| MEMORIZE
Products of powers of sine & cosine, tangent & secant, cotangent and cosecant:
General approach: factor out a potential du, try to convert the rest of the integrand into u
Role of Pythag identities: convert between sines and cosines, or between tangents and secants.
This may help with the Pythagorean relationships:
sec(t) tan(t)
1
Cases for sin & cos:
sin/cos with at least one odd power
Key to approach factor out du=sin(x)dx or du=cos(x)dx
sin/cos with all even powers: key trig identities for cos2x and sin2x: (1±cos 2x)/2
or, reduction formulas, #29, 30
Ex: cos4(x)sin3(x)
sin4(x)
Cases for sec & tan
Key to approach: factor out du=sec2x or du=sec(x)tan(x)
Ex: [Both odd powers]: factor out sec(x)tan(x), use as du, convert the rest to sec(x); u=sec(x)
Three cases:
1. Any odd power of tan(x): factor out one power of tan(x), convert the rest to sec(x), u=sec(x)
2. Any even power of sec(x): factor out sec2(x), convert the rest to tan(x), u=tan(x)
Note: even powers of tan(x) alone can be converted into even powers of sec(x)
3. Otherwise, convert all to an odd power of sec(x), use reduction formula
int(secnu,u) = 1/(n-1) tan(u)secn-1(u) + (n-2)/(n-1) * int(secn-2(u),u)
Handy: int of sec3x is the average of the integral and the derivative of the secant
Derivation: int by parts: the part to integrate is sec2x
Ex: sec3(x)tan(x)
tan4(x)
sec5(x) with the reduction formula
csc and cot are similar: integrating factor for csc(x) is csc(x)-cot(x)
Alt approach for sin(ax)cos(bx) etc. exists (formulas 79-81 in table), but IBP is easier to check by diff
(because you don't change the angles)
Ex: sin(5x)cos(4x)
8.4 Trig Substitution
Steps:
1. identify form:
(a2-x2)n/2: sinq=x/a
(x2-a2)n/2: secq=x/a
(x2+a2)n/2: tanq=x/a
2. draw and label triangle [std position]
3. find
x
dx
value of radical
use Pythag identities,
or read from triangle: what trig func is the radical over the constant?)
4. sub into integral
5. integrate -- using section 8.2 techniques
6. to interpret the integration result, read the trig values out of the triangle.
Ex: Find int(sqrt(4-x^2),x=0..2) -- area inside circle, r=2, in 1st Q [1st use of calculus for area of circle]
Subs: sinq=x/2; results in int(4cos2q), then int(2q+sin2q
Use identity sin2q = 2 sinq cosq
Ex: Find int((4x^2-9)^(-1/2)) and int((4x^2-9)^(-3/2)) [same subs]
subs secq=2x/3
App: Arc length func along parabola y = x^2: s = int(sqrt(1+4x^2),x=0..t) [arc length covered in 9.1]
Subs tanq = 2x/1, results in int of sec3q
Ex: Find int ((x2-4x-1)-3/2,x) to show completing square. Subs: sec q = (x-2)/sqrt(5)
To integrate, convert to sines and cosines.
Completing the square will put any quadratic expression into the forms used in this lesson
8.5 Integrating Rational Functions by Partial Fractions
(For #41, you can factor x^4+1: set x^4+1 = (x^2+ax+1)(x^2-ax+1),
mult out the right side and determine what "a" has to be. On the other hand, you can use u-substitution)
1. Divide first, if degree of num ³ degree of denom
2. Set up
Linear factors Show setup, denom x2+2x-15
Repeated linear factors Show setup: denom: (x+5)(x-3) 3
Quadratic factors (no real roots) Show setup: denom: (x2+5)
Repeated quadratic factors Show setup: denom: (x2+5) 2
Show combination for: (x3+2x-4) / [x*(x2+4) 2] Note: 5th degree denom
3. Add fractions and equate numerators
Full examples:
Find int (1/ (x2+2x-15) , x) Linear, none repeated
Find int ((x-3) / [(x+1)(x2+2)], x)
By equating coeff
By subst of constants for x (use -1, 0, 1)
I recommend using a graphing calculator to solve the system of equations.
For TI models, you can enter the matrix this way:
[[2,3,6,4][1,-3,6,1][6,5,-8,2]]
Evaluate this matrix with the rref function. You can use rref(Ans)
Maple command for partial fraction decomposition: Store your fraction as f:=.... Then,
convert(f,parfrac);
The TI-89 has a function on the algebra menu that will do a partial fraction decomposition.
You should experiment to find which function does the trick.
8.6 Integration Using Tables and Computer Algebra Systems
Do not do the CAS portion for problems 26-89
Keys for tables:
Use u-sub (or even completing the square) to put the integral into the form in the table.
Pay close attention to the conversion between dx and du
The table entry must fit your intrgral exactly.
Be very careful with the substitutions.
Ex: int(sqrt(1+2e^(2x))
Ex: Reduction formula for integral of cot6x
Example 6 and previous discussion: Weierstrass subs for rational functions of sines and cosines:
If t = tan (x/2), then cos (x/2) = 1/sqrt(1+t2) and sin (x/2) = t/sqrt(1+t2)
Therefore sin x = 2t/(1+t2), cos(x) = (1-t2) / (1+t2), dx = 2 / (1+t2) dt
Result is a rational function in t
8.7 Approximate Integration
Source of formulas
Areas of rectangles: A = Δx * y
Areas of trapezoids: A = Δx/2 * (y1 − y0)
Areas under parabolas: A = Δx/3 * (y0 + 4y1 + y2)
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integral of exp(-x^2) from 0 to 3 with 10 increments |
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TRAP |
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MIDPOINT |
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SIMPSON |
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x |
f(x) |
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multiplier |
product |
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multiplier |
product |
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multiplier |
product |
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0 |
1 |
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1 |
1 |
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1 |
1 |
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0.3 |
0.913931 |
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2 |
1.8278624 |
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1 |
0.913931 |
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4 |
3.6557247 |
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0.6 |
0.697676 |
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2 |
1.3953527 |
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2 |
1.3953527 |
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0.9 |
0.444858 |
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2 |
0.8897161 |
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1 |
0.444858 |
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4 |
1.7794323 |
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1.2 |
0.236928 |
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2 |
0.4738555 |
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2 |
0.4738555 |
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1.5 |
0.105399 |
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2 |
0.2107984 |
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1 |
0.105399 |
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4 |
0.4215969 |
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1.8 |
0.039164 |
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2 |
0.0783278 |
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2 |
0.0783278 |
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2.1 |
0.012155 |
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2 |
0.0243104 |
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1 |
0.012155 |
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4 |
0.0486207 |
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2.4 |
0.003151 |
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2 |
0.0063022 |
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2 |
0.0063022 |
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2.7 |
0.000682 |
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2 |
0.0013647 |
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1 |
0.000682 |
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4 |
0.0027293 |
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3 |
0.000123 |
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1 |
0.0001234 |
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1 |
0.0001234 |
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sum: |
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5.9080136 |
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1.477026 |
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8.8620655 |
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n |
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n=10 |
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n=5 |
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N=10 |
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*(b-a)/2n |
0.15 |
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*(b-a)/n |
0.6 |
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*(b-a)/3N |
0.1 |
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integral |
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0.886202 |
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0.886216 |
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0.8862066 |
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actual (from calculator) |
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0.8862073 | ||