formula_sheet - Math 1005 D Formula Sheet Formula Sheet This is only for your reference Please do not bring this to the exam 1 Integral 1.1 Integration

formula_sheet - Math 1005 D Formula Sheet Formula Sheet...

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DRAFT Math 1005 D Formula Sheet Formula Sheet This is only for your reference. Please do not bring this to the exam. 1 Integral 1.1 Integration by parts The technique known as integration by parts is used to integrate a product of two functions, for example Z e x sin(2 x ) dx, Z x 4 e 3 x dx The integration by parts formula: Z u dv dx dx = uv - Z v du dx dx (1.1) Note that the formula replaces one integral, the one on the left, with a different integral, that on the right. The intention is that the latter is simpler to evaluate. Note also that to apply the formula we must let one function in the product equal u . We must be able to differentiate this function to find du dx . We let the other function in the product equal dv dx . We must be able to integrate this function, to find v . 1.1.1 Tabular Integration by Parts The technique of tabular integration allows one to perform successive integrations by parts on integrals of the form Z u ( x ) v ( x ) dx (1.2) without becoming bogged down in tedious algebraic details. There are several ways to illustrate this method, one of which is diagrammed in Table 1. (We assume throughout that u and v are smooth enough to allow repeated differentation and integration, respectively.) Page 1 of 5

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