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Constant of Integration (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Finding the constant of integration

How can I find the value of a constant of integration?

  • When finding an indefinite integral, a constant of integration is needed

    • integral f open parentheses x close parentheses space d x equals F open parentheses x close parentheses plus C

      • where F to the power of apostrophe open parentheses x close parentheses equals f open parentheses x close parentheses

      • and C is any constant

  • If you know more information about F open parentheses x close parentheses you can work out the value of the constant C

    • You may be given the value of F open parentheses x close parentheses for some particular value of x

    • Or you may be told that the graph of F open parentheses x close parentheses goes through a particular point open parentheses x subscript 0 comma space y subscript 0 close parentheses

      • Remember in this case that space y subscript 0 equals F open parentheses x subscript 0 close parentheses

    • This lets you set up and solve an equation to find the value of C

  • For example, if the function F is defined by F open parentheses x close parentheses equals integral open parentheses 2 x plus 3 close parentheses space d x, and you also know that the graph of F goes through the point open parentheses 1 comma space 2 close parentheses

    • First integrate

      • F open parentheses x close parentheses equals integral open parentheses 2 x plus 3 close parentheses space d x equals x squared plus 3 x plus C

    • The graph goes through open parentheses 1 comma space 2 close parentheses, so F open parentheses 1 close parentheses equals 2

      • F open parentheses 1 close parentheses equals open parentheses 1 close parentheses squared plus 3 open parentheses 1 close parentheses plus C equals 2

    • Solve the equation for C

      • table row cell 1 plus 3 plus C end cell equals cell 2 space space rightwards double arrow space space C plus 4 equals 2 space space rightwards double arrow space space C equals negative 2 end cell end table

    • Therefore

      • F open parentheses x close parentheses equals x squared plus 3 x minus 2

  • Finding the constant of integration in this way is equivalent to finding the particular solution of a first-order differential equation in the form fraction numerator d y over denominator d x end fraction equals f open parentheses x close parentheses

    • See the 'Particular Solutions' study guide for a more formal treatment of this

Worked Example

The function h is being used to measure the height of a projectile above the ground at time t. The height h open parentheses t close parentheses is measured in feet, and t is measured in seconds.

It is known that h satisfies the equation h open parentheses t close parentheses equals integral open parentheses 70 minus 32 t close parentheses space d t, and that at time t equals 2 the projectile is 81 feet above the ground.

Find an explicit expression for h in terms of t.

Answer:

First find the indefinite integral; don't forget the constant of integration

h open parentheses t close parentheses equals integral open parentheses 70 minus 32 t close parentheses space d t equals 70 t minus 16 t squared plus C

We know that h open parentheses 2 close parentheses equals 81

70 open parentheses 2 close parentheses minus 16 open parentheses 2 close parentheses squared plus C equals 81

Solve for C

table row cell 140 minus 64 plus C end cell equals 81 row cell C plus 76 end cell equals 81 row C equals 5 end table

Substitute that value for C into the expression for h open parentheses t close parentheses

h open parentheses t close parentheses equals 70 t minus 16 t squared plus 5

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.