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FSU - MAC 2233 - MAC 2233 (Business Calc) Exam 3 Study Guide - Study

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FSU - MAC 2233 - MAC 2233 (Business Calc) Exam 3 Study Guide - Study

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background image MAC 2233: Business Calculus: Exam 3: Study Guide    Francesca Tabana Section 6.2: Integration by Substitution Indefinite Integration Rules Brute Force Technique This technique is used when the integrand is a single transcendental  function or a product of an algebraic function and a transcendental function. Transcendental function: a function that "transcends" algebra in that it 
cannot be expressed in terms of a finite sequence of the algebraic operations
of addition, multiplication, and root extraction. Examples of transcendental 
functions include the exponential function, the logarithm, and the 
trigonometric functions.
Examples Find ∫ 6x 2  dx We can move the 6 outside the integral: ∫ 6x 2  dx = 6 ∫ x 2  dx And now use the Power Rule on x 2 : = 6  3 3   + C Simplify: = 2x 3  + C Find ∫ x (4x 2 +1) dx
background image First expand the part of the integrand being squared: ∫ x (16x 4 +8x 2 +1)   dx Multiply the x in front of the parentheses with each term in the parentheses: 
∫ (16x
5 +8x 3 +x)   dx Now use the Power Rule on each term: 16  6 6  +8   4 4  +  x2 2  + C Clean up the equation by combing like terms:  16 6  x + 2x 4  +  1
2
 x 2  + C → 8
3  x
+ 2x 4  +  1
2  x
2  + C Substitution Technique In this method, the inside function of the composition is usually  replaced by a single variable (often u). The purpose in using the substitution 
technique is to rewrite the integration problem in terms of the new variable 
so that one or more of the basic integration formulas can then be 
applied. This means that we have to omit the variable x while we solve the 
problem, so that the end result will make sense.
**Remember: when choosing ‘u’, you want to pick the more complicated expression;  the polynomial to the highest degree is your safest bet. Examples Evaluate ∫ x (x 3 +1) dx  Because the inside function of the composition is x 3  + 1, substitute as followed: **(You can leave du equal to 3x dx if you want, and then cancel out the x 2 ’s later) Then you plug in place of the (x 3  + 1) and use the power rule to finish up:
background image Evaluate ∫  x ln ¿ ¿ x ¿ ¿ 1 ¿  dx **NOTE: If you ever see an integral in the forms ∫  x ln ¿ ¿ ¿ p x ¿ ¿ ¿  dx  OR  ∫  x ln ¿ ¿ x ¿ ¿ 1 ¿  dx, then the technique you are to use is ALWAYS substitution, and ‘u’ is ALWAYS to equal ‘ln x’. ***With this being said, you will always end up with x du = dx.*** ∫  u ¿ ¿ x ¿ ¿ 1 ¿   x  du <-------------- Substitute ∫  1 u 3  du <------------ The x’s cancel out Bring up the u 3  and use the power rule: ∫ u -3  du =   u 2 2  + C Get rid of the negative exponents:  1 u 2  + C
background image Replace the original expression for u so the answer is in terms of x:  x ln ¿ ¿ 2 ¿ ¿ 1 ¿  + C Section 6.5: Definite Integrals Components , where  a = the lower limit/boundary and b = the upper  limit/boundary. **Remember: the upper limit does NOT have to be greater than the lower 
limit.
**When solving definite integrals, ALWAYS evaluate the upper limit first!! Definite Integration Rules Example Evaluate  0 4  dx First, use the power rule: ( 8 x 2 2  + C) -----> 4x 2  + C Now, evaluate from the limits: 4(4) 2  + C – (4(0) 2  + C)

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School: Florida State University
Department: Calculus and Pre Calculus
Course: Business Calculus
Professor: Kenneth Doraro
Term: Summer 2016
Tags: businesscalculus, Calculus, and business
Name: MAC 2233 (Business Calc) Exam 3 Study Guide
Description: This EXTENSIVE study guide covers topics 6.2, 6.5, 7.1, 7.2, 8.1, and 8.2. Examples from class are included, with the steps shown as well!
Uploaded: 12/03/2017
16 Pages 186 Views 148 Unlocks
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