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**Unformatted text preview: **6.2 Integration by Substitution & Separable Differential Equations M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002 The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate. Example 1: ( 29 5 2 x dx + Let 2 u x = + du dx = 5 u du 6 1 6 u C + ( 29 6 2 6 x C + + The variable of integration must match the variable in the expression. Dont forget to substitute the value for u back into the problem! Example: (Exploration 1 in the book) 2 1 2 x x dx + One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is . 2 1 x + 2 x dx 1 2 u du 3 2 2 3 u C + ( 29 3 2 2 2 1 3 x C + + 2 Let 1 u x = + 2 du x dx = Note that this only worked because of the 2 x in the original....

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