1. Unit 4:Mathematics Introduce various integration functions Aims
Objectives
Understand the process involved in integration as reverse differentiating, indefinite integrals, constant of integration, sines and cosines , logs and exponentials

2. Calculus Calculus falls into two sections: Differentiation Integration
The second major component of the Calculus is called integration. This may be introduced as a means of finding areas using summation and limits.

3. Antiderivatives— differentiation in reverse.
by differentiating term 3x² + 7x − 2.
f(x) =dF = 6x + 7. dx
Suppose now that we work back to front and ask ourselves which function or functions could possibly have 6x + 7 as a derivative.

4. Integration as reverse of differentiating
We say that
F(x) = 3x² + 7x − 2
is an antiderivative of
f(x) = 6x + 7.
There are however other functions which have derivative 6x + 7. Some of these are
3x² + 7x + 3, 3x² + 7x,
3x² + 7x − 11
If F(x) is an antiderivative of f(x) then so too is F(x) + C for any constant C.

6. To integrate a term, increase its power by 1 and divide by this figure
To integrate a term, increase its power by 1 and divide by this figure. In other words:

7. y = x3 is ONE antiderivative of
There are infinitely many other antiderivatives which would also work:
y = x3 + 4
y = x3 + π
y = x
In general: y = x3 + C, is the indefinite integral. C is called the constant of integration.

9. Integrate . From exponential functions and properties of integrals we get that
= x ln(x) - x + C.
ln(x) dx

10. 5x + C
2.x C 9
3. −1 + C x
e3x + C 3
5. −1 e−2x + C 2
6. 2 sin 1 x + C.

12. Area under a curve Suppose we wish to evaluate
This limit of a sum arises when we use the sum of small rectangles to find the area under y = x² between x = 0 and x = 1. This limit defines the definite integral
Note that the C’s cancel