2. HIGHER MATHEMATICS Unit 2 - Outcome 2 Integration

3. Differential Equations From the chapter on differentiation, we know that Ify = x 2 Then dy dx =2x But what if we already knew the derivative, and wanted to work out y?

4. dy dx =2x y = ? y = x 2 y = x 2 + 1 y = x 2 + 2 y = x 2 + 3 y = x 2 + 4 y = x 2 + 5 y = x 2 + 6 y = x 2 - 1 y = x 2 - 2 y = x 2 - 4 y = x 2 - 5 y = x 2 - 7 y = x 2 + 7 y = x 2 + 8 y = x 2 + 11 y = x 2 + 47 y = x 2 - 19 y = x 2 + 71 y = x 2 - 88 y = x 2 + 34 y = x 2 + 40 y = x 2 - 75 y = x 2 – 9.3 y = x 2 + 789 y = x 2 - 400 There are infinitely many solutions!

5. We write the solution as follows : y = x 2 + c This is called the GENERAL SOLUTION to the DIFFERENTIAL EQUATION To find the value of c, we would need information about a point which lies on the curve.

6. EXAMPLE 1 A curve is defined such that dy dx =2x If the curve passes through the point ( 1,5 ), express y in terms of x. dy dx =2x y = x 2 + c Sub. (1, 5) y = x 2 + c5 = 1 2 + c c = 4  y = x 2 + 4

7. EXAMPLE 2 Given that dy dx =6x 2 + 4x + 1, and x = -1 when y = -7, express y in terms of x. dy dx =6x 2 + 4x + 1 y = 2x 3 + c Sub. (-1, -7) + 2x 2 + x -7 = -2 + 2 - 1 + c c = -6  y = 2x 3 + 2x 2 + x - 6

8. dy dx =…. SUMMARY is called a DIFFERENTIAL EQUATION y = x 2 + c is called the GENERAL SOLUTION y = x 2 + 4 is called the PARTICULAR SOLUTION

9. Integration From the chapter on Differentiation, we know that …….. d dx (x n )=nx n-1 d dx (x n+1 )=(n+1)x n d dx =xnxn x n+1 () differentiationanti-differentiation n + 1

10. The process of finding the ANTI- DERIVATIVE is called INTEGRATION. This is how it is written. ax n dx= ax n + 1 n + 1 + c I will explain this later. CONSTANT OF INTEGRATION DON’T FORGET TO INCLUDE!

11. EXAMPLES Integrate each of the following with respect to x. 1.6x 2 6x 2 dx 6x 3 3 =+ c = 2x 3 + c 2.12x 3 12x 3 dx 12x 4 4 =+ c = 3x 4 + c

12. 3.8x + 3 (8x + 3) dx = 4x 2 + 3x + c 4.6x 5 + x - 1 (6x 5 + x - 1) dx 8x 2 2 =+ c+ 3x 6x 6 6 =+ c x2x2 2 +- x x6x6 =+ c½x 2 +- x

13. 5.(x - 7)(x - 1) (x - 7)(x - 1) dx x3x3 3 =+ c 8x 2 2 -+ 7x(x 2 - 8x + 7) dx =x3x3 =+ c4x 2 -+ 7x 1 3 6.3x(x 2 – 5) 3x(x 2 – 5)dx 3x 4 4 =+ c 15x 2 2 -(3x 3 – 15x) dx = x4x4 =+ c+ 3 4 x2x2 15 2