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Unformatted text preview: This section extends your antiderivative finding skills dramatically with a technique called substitution. Essentially, if a function's antiderivative can be found by "backing through" the chain rule, then this technique is useful. To have sucess here, you must have command of all the derivative rules and the antiderivative rules discussed so far. Lots of practice and answer checking is essential. Example: Given sin cos what is cos ? , we know that We know it must be sin We can obtain this antiderivative by substitution, as follows: a antiderivative rule we know is cos sin cos , we let , In the antiderivative then so we substitute to have cos cos sin for to give and then back-substitute sin MTH 173 section 5.5, page 1 One integral rule is that a constant factor can be "moved" across the integral, as We use this to introduce missing constant factors: Example: , so the form of the integral is Example: cos sin sin cos sin cos , so the form of the integral is csc sin MTH 173 section 5.5, page 2 Example: let then Multiply in the integral by , outside by backsubstitute Example: no substitution required MTH 173 section 5.5, page 3 Example: cos cos let then multiply in the integral by , outside by cos cos backsubstitute cos Example: sin cos sin sin let cos , then sin multiply inside and outside by sin cos cos
2 sin back- cos substitute sec cos MTH 173 section 5.5, page 4 Example: sec tan sec tan backsubstitute sec tan sec sec : Now use the given information to determine the value of sec cos the function is: 2. sec , 8. so MTH 173 section 5.5, page 5 14 22 arctan arctan arctan arctan arctan 26 tan tan sec sec tan MTH 173 section 5.5, page 6 32 ln ln 44 Here we use substitution to make the radicand of then replace by solving for : just a variable, so MTH 173 section 5.5, page 7 50. Given In a definite integral that requires substitution, we can change the limits of integration and then not need to back-substitute (tho it's not necessary). Here we use the substutition and the lower limit then we have the upper limit MTH 173 section 5.5, page 8 ...
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