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First year: (I) Basic integration Integration: areas The symbol b f ( x)dx a means “area of the set situated between the graph of f , the x axis, the line of equation x a and the line of equation x b ”. It’s easier to understand this notion through an example: 2 ( x 1)(3 x)dx is the area of the set situated between the x axis, the graph 1 of f and the vertical lines of equation x 1, x 2 . (II) Basic Rules Integration is, intuitively, the opposite of differentiation. Mathematically, we say that if there exists a function F such that d F ( x) f ( x) dx b then f ( x)dx F (b) F (a) . a The result above is sometimes called the fundamental theorem of calculus. When you are asked to integrate a function f , the trick is to find this function F. If such a function F exist, then instead of saying “a function F such that d F ( x) f ( x) ”, we usually say dx F ( x) f ( x)dx Let’s have a look at an example: 2 Calculate: 2xdx . 1 The first step is to find xdx . We know that the derivative of x 2 is 2x , so that we have 2 xdx x It remains now to calculate the actual value of the integral: 2 2 xdx [ x ] 2 2 1 2 3. 2 2 1 1 So far so good? Now try to calculate the following integral: 2 xdx . 1 Here are a few examples: - 3 x dx x4 4 3 - 5dx [5x] 3 0 5 3 5 0 15. 0 - 2 7 2 x8 x8 5 x dx 5 8 20 . The general rule is n x dx x n1 . n 1 2 . (III) Exercises Find the following integrals: - x dx - 3x dx 2 4 3 - x 1 3 dx