## Integration

is the reverse process 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 it is boundless and the value of c is unknown.

In general, the rule for integrating 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.

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.
At this level, only polynomial functions will be integrated.

Expressions with brackets should usually be expanded first.

Examples
Find ∫ x3 dx ∫ x3 dx = + c
Find ∫ 3x2 + 4x + 5 dx
 ∫ 3x2 + 4x + 5 dx = + c = x3 + 2x2 + 5x + c
Find ∫ (x − 2)(x + 4) dx
 ∫ (x − 2)(x + 4) dx = ∫ x2 + 2x − 8 dx = + c = x3/3 + x2 − 8x + c

### Finding the Constant of Integration

Sometimes enough information is given to enable the value of the constant to be found.

Example

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 x 1 + c

8 = 4 + c

c = 4

f(x) = 2x2 + 2x + 4