MA 141 Final: Substitution

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As we saw previously, sometimes integrals represent a function composition. Substitution is a method to deal with such integrals. Recall that if h(x) = f (g(x)) and f = f , then. H (x) = f (g(x)) g (x) = f (g(x)) g (x) Z f (g(x)) g (x)dx = f (g(x)) + c. We can simplify the problem by letting u = g(x). Recall that we said in section 3. 6 if y = f (x), then dy = f (x)dx. Then here we have du = g (x)dx and. Z f (g(x)) g (x)dx = z f (u)du = f (u) + c = f (g(x)) + c. Using this change of variables is called the substitution method. For substitution to work, we pick part of our integrand to be u and ensure that this relabeling with u and du leaves behind none of the old variable (the change must be complete).