Integration Revision 5
Integration is the reverse of differentiation. However: If y = 2x + 3, dy/dx = 2 If y = 2x + 5, dy/dx = 2 If y = 2x, dy/dx = 2 So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc. For this reason, when we integrate, we have to add a constant. So the integral of 2 is 2x + c, where c is a constant. A "S" shaped symbol is used to mean the integral of, and dx is written at the end of the terms to be integrated, meaning "with respect to x". This is the same "dx" that appears in dy/dx . To integrate a term, increase its power by 1 and divide by this figure. In other words:
When you have to integrate a polynomial with more than 1 term, integrate each term. So:
Now we will see how to find a particular solution to an equation. To do this all we need to do is fix the curve to a specific position, and we will have one solution instead of a family of curves. For example if
Integrating, we know that
This gives us a family of curves which all have the same gradient function
If we are told that our curve passes through the point (1, 9) then it can only be one of these curves. If it passes through (1, 9), then
Another example could be:
Here is an exam style question:
When you have to integrate a polynomial with more than 1 term, integrate each term. So:
Now we will see how to find a particular solution to an equation. To do this all we need to do is fix the curve to a specific position, and we will have one solution instead of a family of curves. For example if
Integrating, we know that
This gives us a family of curves which all have the same gradient function
If we are told that our curve passes through the point (1, 9) then it can only be one of these curves. If it passes through (1, 9), then
Another example could be:
Here is an exam style question:
