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First year: (I) Integration by substitution The method Remember that f ( x)dx F ( x) , where dF ( x) f ( x) so, in a way, integration dx is the “opposite” of differentiation. Integration by substitution is in this case the “opposite” of the chain rule for differentiation. 2x dx . Let’s have a look at an example: calculate 2 x 1 We are looking for a function F (x) such that dF ( x) 2x 2 . dx x 1 Now, if we set u ( x) x 2 1 , we get 2x u ' ( x) x 1 u ( x) 2 and we know that only one type of function F can have a derivative of this form: the logarithm dF ( x) u ' ( x) . dx u ( x) Note that we used the chain rule to differentiate F . F ( x) ln( u ( x)) Hence x 2x dx ln( u ( x)) ln( x 2 1). 1 2 Now, “integration by substitution” means that you change the variable in order to transform the function inside the integral into a standard function. Let’s have a look at the example above using this method: Set u x 2 1 . Then du 2x dx du 2 xdx , from which we get x 2x 1 dx du ln( u ) ln( x 2 1). u 2 2 (II) Worked example and exercises (i) Calculate xx 2 1 dx. 6 Set u x 2 1 . Then 1 du xdx , so that 2 7 1 6 1 u7 x2 1 x x 1 dx u du . 2 2 7 14 2 6 (ii) Calculate sin( x)cos( x) 3 dx. (iii) Calculate x2 x 3 1 dx . (iv) Calculate ex 1 2e x 2 x dx. (v) Calculate sin( 2 x) 3 dx. (vi) x (vii) 5 2 x ln( x 2 1) 2 1 x 2e x e cos( 2 x) x3 3 1 dx. 3 dx.