Integration is the reverse of differentiation.
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:
In the above examples, there was always a constant term left over after integrating. For this reason, such integrals are known as indefinite integrals. With definite integrals, we integrate a function between 2 points, and so we can find the precise value of the integral and there is no need for any unknown constant terms.
Finding the area under a curve
The area under a curve can be found be integrating, if the equation of the curve is known.
To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b.
Areas under the x-axis will come out negative and areas above the x-axis will be positive. This means that you have to be careful when finding an area which is partly above and partly below the x-axis.