math.com
Home    |    Teacher    |    Parents    |    Glossary    |    About Us
Homework Help Practice Ask An Expert Calculators & Tools Games Store
Email this page to a friend Email this page to a friend
Resources
· Cool Tools
· Formulas & Tables
· References
· Test Preparation
· Study Tips
· Wonders of Math
 
Search
Custom Search



Math Jobs


  
Proof: Integral bx
(Math | Calculus | Integrals | Table Of | bx)
 
Discussion of
(integral) bx dx = bx / ln(b) + C.
1. Proof
    Strategy: Use the (integral) eu du = eu + C

    since eln b = b,

    (integral) bx dx = (integral) [ (eln b) x ] dx
    (integral) e (ln b) x dx

    set u = (ln b) x
    then du = (ln b) dx
    substitute...

    (integral) eu (du / ln b)
    = (1 / ln b) (integral) eu du

    solve the integral...

    = (1 / ln b) ( eu + C )
    = (1 / ln b) eu + C2   (create new constant)

    substitute back u = (ln b) x,

    = ( 1 / ln b) e(ln b) x + C2
    = ( 1 / ln b) ( e(ln b) )x + C2
    = ( 1 / ln b) bx + C2
    = bx / ln b + C2
    Q.E.D.

    See also the proof of (integral) eu du = euPROOF

2. You need not memorize this theorem.  Derive it each time you use it.
    Consider this example: if you have the integral:
     (integral) 2x dx.
    There is no need to memorize the formula.  We will get this integral into the easier form, (integral)eu du.

    Recall that eln(2) = 2

    (integral) 2x dx = (integral) ( eln (2) ) x dx
    =(integral) eln (2) x dx

    set u = ln(2) x
    then du = ln(2) dx
    substitute:

    (integral) eu (du / ln 2 )
    = (1 / ln 2) (integral)eu du
    = (1 / ln 2) eu + C
    substitute back...
    = (1 / ln 2) eln(2) x + C
    = (1 / ln 2) ( eln(2) )x + C
    = (1 / ln 2) 2x + C  ANSWER

    This method is actually quite fast; it just looks long because I drew it out for demonstration purposes.

  
 
  

 
Contact us | Advertising & Sponsorship | Partnership | Link to us

© 2000-2005 Math.com. All rights reserved.     Legal Notices.     Please read our Privacy Policy.