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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Integration

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Integration is the reverse of differentiation.

If

If

If

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. E.g.:

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:

Definite Integrals

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.

(diagram for example missing)

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 between x = a and x = b, integrate between the limits of a and b.

(diagram for example missing)

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.

(diagram for example missing)

Comments

Several diagrams missing.

The article is only suitable for basics at C1 and not the fractional and negative powers of x and other basic functions needed for C2 onwards.

More needed explaining what definite integrals are and why we might find then (ie the area under a curve section linked in).