Advertisement
Higher Bitesize
Print

Maths

Integration

Page:

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  1. Next

Indefinite integration

Integration is the inverse process to differentiation. So instead of multiplying by the index and reducing it by one, we increase the index by one and divide by the new index. For example, x^2 becomes {{x^3 } \over 3} + c.

The + \;c appears because the derivative of any constant term is zero.

c is called the (arbitrary) constant of integration. Its value can be found when appropriate additional information is provided, and this gives a particular integral.

The rule for integration is

f'(x) = ax^n \Rightarrow f(x) = {{ax^{n + 1} } \over {n + 1}} + c

provided n \ne -1.

This can also be written in the form \int {ax^n dx = {{ax^{n + 1} } \over {n + 1}} + c} provided n \ne -1.

In general \int {f'(x)dx = f(x) + c} or \int {({{dy} \over {dx}})dx = y + c}

Here's an example.

Find the equation of the curve for which {{dy} \over {dx}} = 4x^3 + 6x^2 and which passes through the point (1, 3).

integrating gives

y = \int ( 4x^3 + 6x^2 )dx = x^4 + 2x^3 + c

substituting x = 1 and y = 3 gives 3 = 1 + 1 + c therefore c = 1

y = x^4 + 2x^3 + 1

Question

Find the equation of the curve for which {{dy} \over {dx}} = 2x + 1 and which passes through the point (2, 9).

toggle answer

Answer

integrating

y = \int {(2x + 1)dx = x^2 + x + c} substituting x = 2 and y = 9 \eqalign{ \Rightarrow 9 & = & 4 + 2 + c \cr \Rightarrow c & = & 3 \cr \Rightarrow y & = & x^2 + x + 3 \cr}

Page:

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  1. Next

Back to Calculus index

Links

LTS - Mathematics

Download some of LTS'mathematic resources, have a look at the Sunday Herald's Exam Guide or brush up on your study skills.

Links

On the web

Explore the BBC

This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.