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


  
Proof: Integral sinh(x)
(Math | Calculus | Integrals | Table Of | sinh x)
 
Discussion of
(integral) sinh x dx = cosh x + C.

1. Proof

    Strategy: Use definition of sinh.
     
    sinh x =
    ex - e-x
    2
    (integral) sinh x dx = (integral)
    ex - ex
    2
    dx
    = (1/2) (integral) ex dx - (1/2) (integral) ex dx
    solve left equation
    = (1/2) ex - (1/2) (integral) ex dx

    set
      u = - x
    then we find
      du = - dx

    substitute du= - dx, u= - x

    = (1/2) ex + (1/2) (integral) - ex dx
    = (1/2) ex + (1/2) (integral) eu du

    solve the right integral
    = (1/2) ex + (1/2) eu + C

    substitute back u= - x
    = (1/2) ex + (1/2) ex + C
    ex + ex
    2
    + C
    which by definition
    = cosh x + C
    Q.E.D.

  
 
  

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

© 2000-2023 Math.com. All rights reserved.     Legal Notices.     Please read our Privacy Policy.