Integrals By Zac Cockman Liz Mooney. Integration Techniques Integration is the process of finding an...
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Transcript of Integrals By Zac Cockman Liz Mooney. Integration Techniques Integration is the process of finding an...
Integrals
ByZac CockmanLiz Mooney
Integration Techniques
• Integration is the process of finding an indefinite or diefinite integral
• Integral is the definite integral is the fundamental concept of the integral calculus. It is written as
• Where f(x) is the integrand, a and b are the lower and upper limits of integration, and x is the variable of integration.
Integration techniques
• Integration is the opposite of Differentiation.
• Power Rule• U-Substitution• Special Cases• Sin and Cos
Power Rule
•
• n cannot equal -1• u=x• Du=dx• N=1
• C = constant
+ c
Examples
U = 2xDu = dxn = 1
Answer = X2 + C
Examples
• Answer = X3/3 + 3x2/2 + 2x + C
U=x U=x U=x
Du=dx Du=dx Du=dx
N=2 N=1 N=0
Examples
• U = 4 X2
• Du = 8xdx• N = -1/2
• 3/8 * 2 * (4x2 + 5)1/2 + C• Answer ¾(4X2 + 5)1/2 + C
Try Me
Try Me
•
Try Me Continue
• U = 1 +x2
• Du = 2dx• N = -1/2• 1/2 [2(1+x2)1/2] + C• (1+x2)1/2 + C
Try Me
Try Me
• U = x4 + 3• Du = 4x3 dx• N = 2
Try Me Continued
U-Sub
• What is U-Sub• When do you use it• Steps• Find your u, du, and for u, solve for x• Replace all the x for u. • Do the same steps for power rule• At the end replace the u in the problem for
your u when you found it in the beginning.
Example
• U=• X= u2 -1• dx= 2udu• (u2 – 1) u(2udu)• 2u4 – 2u2
• 2/5 (u5 – 2/3u3) + c• 2/5 (x+1) 5/2 + c
Example
• U =• U2 – 1 = x• 2udu = dx
2/3(x+1)3/2 -2(x+1) + c
Try Me
Try Me
U =
X =
Du = udu
1/10 u5 + 1/2u2 + c
1/10 (2x-3)5/2 + ½(2x-3) + c
Special Cases
• When n = -1 the u is put inside the absolute value of the natural log
• If there is only one x in the problem and it is squared, square the term before taking the interval
Special Cases
• Examples
• U = x-1• Du = dx• N = -1
Special Cases
• Examples
Integration using Powers of Sin and Cos
• Three Methods– Odd-Even Odd-Odd Even-Even
• In Odd-Even, take the odd power and re write the odd power as odd even
• Re write the even power change it using Pythagorean identity.
• In Odd-Odd, take one of the odds, change to odd even
• Use same rules
Integration using Powers of sin and cos
• For Even-Even, change the power to the half angle formula.
Special CaseIf the Power of the trig is 1, u is the angle
Powers of Trig Odd - Even
• Take the odd power, re write the odd power as odd
even
• Re write the even power, change it using the
Pythagorean identities.
• ∫sin5xcos4xdx
• ∫sin4x sinxcos4xdx
• ∫(1-cos2x)2 sinxcos2xdx
Powers of Trig Odd-Even
• ∫(1-2cos2x+cos4x) sinxcos4xdx
• ∫sinxcos4xdx-2 ∫sinx cos6xdx+∫sinxcos8xdxU = cosx U = cosx U = cosx
Du = -sinx Du = -sinx Du = -sinx
N = 4 N = 6 N = 8
-1/5cos5x+2/7cos7x-1/9cos9x+c
Try Me
∫sin32xcos22xdx
Powers of Trig Odd - Even
• Try Me• ∫sin32xcos22xdx• ∫sin22xsin2xcos22xdx• ∫(1-cos22x) sin2xcos22xdx• -1/2 ∫sin22xcos22xdx+1/2 ∫sin2xcos42xdx
U = cosx U = cosx
Du = -2sin2x Du = -2sin2x
N = 2 N = 4
-1/6 cos32x+1/10cos52x+c
Powers of Trig Odd Odd
• Take one of the odds, change to odd even. Use other rules to finish.
• Example
Powers of trig Odd-Odd
U = cosx U = cosx
du= -sinxdx du= -sinxdx
n = 3 n = 5
Powers of Trig Odd-Odd
• Example
Powers of Trig Odd Odd
• Example Continued
U = cosx U = cosx
Du = -sinx Du = -sinx
N = 17 N = 19
Powers of Trig Even-Even
• Change to half angle formula• ∫sin2xdx• ∫1-cos2xdx• 2• 1/2∫dx-(1/2)(1/2)∫2cos2xdx
U = x U = 2x
Du = dx Du = 2dx
1/2x-1/4sin2x+c
Try Me
∫sin2xcos2x
Powers of Trig Even-Even
• Try Me• ∫sin2xcos2x• ∫(1-cos2x)(1+cos2x)
2 2• 1/4∫(1-cos22x)dx• 1/4∫sin22xdx• 1/4∫1-cos4x/2dx• 1/8∫dx-(1/4)(1/8) ∫4cos4xdx• 1/8x-1/32sin4x+c
Solving for Integrals
• U =x-1• Du = dx• N = 2
• 9 – 0 = 9
Try Me
• Try Me
Solving for Integrals
• U = x2 + 2• Du = 2xdx• N = 2
Bibliography
• www.musopen.com• Mathematics Dictionary, Fourth Edition,
James/James, Van Nostrand Reinnhold Company Inc., 1976