Rules for Integration

Scalar Multiple Rule:

The integral of a constant times a function is the constant times the integral of the function.

Sum Rule:

The integral of the sum of two functions is the sum of their separate integrals.

Power Rule:
, where 
The integral of a variable raised to a power, n, is the variable to the power (n + 1) over (n + 1), plus a constant.
 

, where and C is a constant.

 Fundamental Theorem of Calculus: If f(x) is continuous on the interval [a,b], then

where F(x) is any function such that F'(x) = f(x) for all x in [a,b].

Mean Value Theorem for Integrals:

 

In the diagram, the area of the rectangle is the same as the area under the curve of f(x) between a and b.

The area of the rectangle is its height times its width:

.

The area under the curve is .

The Mean Value Theorem for Integrals states that there is a value c such that these two areas are equal. The height of the rectangle f(c) is called the average value of f(x) between a and b.

Consequently,

With this technique we can sometimes replace one integral with another, possibly easier, integral.