Math_137_Winter_2010_Week_12_Notes

Math_137_Winter_2010_Week_12_Notes - Math 137 Week 12 Notes...

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Math 137 Week 12 Notes – Dr. Paula Smith Section 5.5 Substitution or Change of Variables So far, the only way we know to find an antiderivative for a function f is to recognize f as the derivative of some function F. Substitution will enable us to expand our repertoire of integrable functions by turning integrals we don’t recognize into ones we do. Substitution is the Chain Rule, backward. Recall that [f(g(x))] = f (g) g (x). Then f(g(x)) = [f(g(x))] dx = f (g) g (x) dx = f (g) dg . We want to find an outer function f with an inner function g(x) such that the rest of the integrand forms g (x) dx; then we can solve the easy integral of f (g) dg. Substitution with Indefinite Integrals Direct Method: By personal preference, I use z instead of g. Look for z(x) (or z of another original variable): In the denominator Under the radical Up in the exponent Within parentheses As a repeated expression As the argument of a function Then take dz, and see if it (or a constant multiple of it) is also in the integrand. Substitute z and
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Math_137_Winter_2010_Week_12_Notes - Math 137 Week 12 Notes...

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