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

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.

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.

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.

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:

y = x3 is ONE antiderivative of There are infinitely many other antiderivatives which would also work: y = x3 + 4 y = x3 + π y = x3 + 27.3 In general: y = x3 + C, is the indefinite integral. C is called the constant of integration.

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

5x + C 2.x9 + C 9 3. −1 + C x 4. 1 e3x + C 3 5. −1 e−2x + C 2 6. 2 sin 1 x + C.

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