4.1_B_Lecture Notes - AP BC Calc 4.1(contd Integration by...

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AP BC Calc 4.1 (cont’d) Integration by Substitution and Trigonometric Substitution Ferguson Notes For integrals that don’t have a specific formula that you can just remember, there are a couple of methods that you can go about using: the first two are somewhat simple, and require little less than algebraic substitution and remembering integration rules. Integration by SubstitutionThe first method that can be used if there is no specific formula for an integral is the Substitution Rule. For instance, consider the integral 221xx dx. How do we approach this? We know how to do an integral of a root using the power rule, so is there some way that we can “change” this integral without altering its value? Sure..let’s do it using a combination of algebra and differentiation. Let 21ux. If we let this be true, then the differential of u is 2duxdx. Notice that the term 2xdxis in the original integral, so we can make a substitution! 222112xx dxxx dxu du. Aha! Doing this substitution makes the integral much, much easier to perform. 3/21/23/22332uu duuduCuC. Now, at this point we’re done with the actual integration, but since we’re asked to find the integral in terms of x, we have to make that final substitution using 21ux3/22222113xx dxxC. In general, this method works whenever we have an integral that we can write in the form ( )( )fg xgx dx. In fact, if Ff , then ( )( )( )Fg xgx dxF g xCBasically, the Substitution Rule is a method used in reverse of the Chain Rule for Derivatives. The Substitution Rule (Official Definition) If ( )ug xis a differentiable function whose range is an interval Iand fis continuous on I, then ( )( )( )fg xgx dx
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