Integration means finding the area between the graph of a function and the x-axis.
The process of integration can be thought of as the reverse of differentiation. It is also known as anti-differentiation.
Given a derived function, integrating will revert back to the original function.
The result of integration is called the integral or anti-derivative.
e.g If x2 is differentiated the result is 2x
The anti-derivative of 2x is x2.
The problem is that many other functions differentiate to 2x. e.g. x2 + 6, x2 − 20 etc.
To allow for this possibility a constant of integration(c) or arbitrary constant has to be added each time we integrate.
This type of integral is called an indefinite integral because the value of c is unknown.
The definite integral is a number, the indefinite is a function.
In general, the rule for integrating powers of x (with the exception of x -1) is "add one to the exponent and divide by the new exponent."
The symbol used to indicate integration is ∫
e.g ∫ 2x dx means "integrate 2x with respect to x" The dx part is simply to indicate that x is the variable.
Methods of Integration
The formula for integration is:
∫ xn dx = + c
For a term with a coefficient not equal to 1, the formula becomes:
∫ axn dx =a. + c
This means that a coefficient remains unchanged by integration.
Functions containing all types of terms and expressions can be integrated. e.g. Trigonometric, logarithmic, roots etc.
Anti-differentiating can be explained by asking "What must I differentiate to get this function"
|(see Example 2)|
|(see Example 3)|
|(see Example 4)|
|1||Find ∫ x3 dx||∫ x3 dx = + c|
|2||Find ∫ (3x2 + 4x + 5) dx||
|3||Find ∫ (x − 2)(x + 4) dx||
Finding the Constant of Integration
Sometimes enough information is given to enable the value of the constant to be found.
Anti-differentiate to help find f(x) if f '(x) = 4x + 2 and f(1) = 8
∫ 4x + 2 dx = 2x2 + 2x + c
i.e. f(x) = 2x2 + 2x + c
8 = 2 x 12 + 2 × 1 + c
8 = 4 + c
c = 4
f(x) = 2x2 + 2x + 4