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Integration by Parts

# Integration by Parts - Integration by Parts Integration by...

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Integration by Parts Integration by Parts is the name of a technique for integrating certain types if functions. It is based on the Product Rule for differentiation, namely ( 29 uv uv u v = + . Integrating this expression (assuming the appropriate antiderivatives exist) yields uv uv dx u vdx = + udv vdu = + . Writing this last expression in a different order yields the formula udv uv vdu = - . In applying this formula keep the following things in mind: One must identify the choices of u and v . Some choices may work and others may not. For the definite integral, the formula is ( 29 b b b a a a udv uv vdu = - . Choosing u and v is a guess. You get better at making successful guesses by gaining experience through working problems. The examples below illustrate these ideas. Example 1 : Evaluate ln x xdx . Solution : Choose ln , u x dv xdx = = . Then 2 1 , 2 dx du v x x = = . ( We usually write this as 2 ln , 1 , 2 u x dv xdx dx du v x x = = = = . Notice that ln x xdx udv = .) Then ln x xdx udv uv vdu = = - 2 2 1 1 ln 2 2 dx x x x x  = -   2 1 1 ln 2 2 x x xdx = - 2 2 1 1 1 ln 2 2 2 x x x C = - + ( 29 2 1 2ln 1 4 x x C = - + . Example 2 : Evaluate 2 sec t tdt Solution : Let 2 , sec , tan u t dv tdt du dt v t = = = = .

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Integration by Parts - Integration by Parts Integration by...

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