Student Xpress Homepage | CSPE | Educational Supplement | Career Guidance | Student Articles | Features

Integration I

Education Links

Leaving Cert

Maths
French
English
Chemistry
Physics
Biology
Economics
Spanish
Geography
History


Junior Cert

Science






 1. The Idea: Integration is the opposite to differentiation. It could be called anti-differentiation.

Notation:

Think of a function which, when you differentiate it with respect to x, you get f(x).
Note: f(x) is called the integrand.

2. Rules of differentiation
(i) Sum Rule:
You can integrate each function of a sum of functions separately.
(ii) Multiplication by a constant rule:
You can take constants outside integrals.

3. Technique:
The most important technique on the core course is substitution. It is used for products and quotients of functions which cannot be simplified into a sum of simple functions.

Let u = that part of the integrand such that when you differentiate it you get a multiple of another part of the integrand.

4. Types of Integrals of the LC Course:
[A] Alint: Integrals with pure algebraic functions.
[B] Expint: Integrals with any hint of the exponential function.
[C] Logint: The inverse linear function.
[D] Trigint: Integrals with pure trig functions.
[E] Specials: Three algebraic functions that require unusual substitutions.

[A] Alint
The basis of Alint is for all
except when p = -1 when
In words: Add one to the power and divide by the new power for all powers except -1 when the answer is ln x.
Note 1: could be regarded as the odd man out (OMO). Keep an eye out for it.
Note 2:

Types of ALINT:
Type 1 - Straights: Sums of multiples of
Steps
  • Tidy up the algebra (multiply out brackets etc..)
  • Integrate out term by term.

    Example 1

    Evaluate
    Solution

    Example 2

    Evaluate
    Solution
    Type 2 - Products and quotients of Algebraic functions which cannot be simplified into sums of multiples of

    Steps
  • Tidy up the algebra.
  • Let u = the function inside the complicated bracket.

    Example 3

    Evaluate
    Solution

    Example 4

    Evaluate
    Solution

    Example 5

    Evaluate
    Solution

    Example 6

    Evaluate
    Solution
    [B] Expint
    The basis of Expint is . This is the easiest function on the course to integrate because when you integrate it you get the same function back again. It can be extended:


    Types of EXPINT:
    Type 1 - Straights: Expofunctions which can be simplified into sums of expofunctions of the form .

    Steps
  • Tidy up the expoalgebra.
  • Integrate out term by term using

    Example 7

    Evaluate
    Solution

    Example 8

    Evaluate
    Solution
    Type 2 - Products and Quotients of Expo and non-expo functions.

    Steps
  • Tidy up the algebra.
  • Let u = the power of the expofunction.

    Example 9

    Evaluate
    Solution

    Example 10

    Evaluate
    Solution
    Type 3 - Products and quotients of pure expofunctions which cannot be simplified into a sum of expofunctions of the form .

    Steps
  • Tidy up the algebra.
  • Let u = the expofunction inside the complicated bracket.

    Example 11

    Evaluate
    Solution

    Example 12

    Evaluate
    Solution
    [C] Logint
    This is the integration of the inverse linear function. It is based on

    However, this can be extended to

    Example 13

    Evaluate
    Solution

    Back to Maths Homepage | Prev | Next












    • Student Xpress Homepage | CSPE | Educational Supplement | Career Guidance | Student Articles | Features