A standard substitution

This week, we looked at the method of substitution for calculating integrals. Recall that the idea of substitution is to "reverse" the chain rule in order to simplify integrals of certain products.

For instance, to calculate , you notice first, that the integrand is a product, and that there are two factors: a simpler one, , and a more complicated one, . Moreover, the simple one is (a constant times) the derivative of part of the complicated one: the derivative of is . So we let , and calculate its differential: . In the integral, we can then substitute u for in the sine function, and for we can substitute . The result is that = , after "un-substituting" for u .

The standard substitution: The purpose of this note is to highlight a specific kind of substitution that comes up so often that it should become "second nature" to you. It occurs when you are trying to integrate a function (like exponential, sine or cosine, logarithm, square root, etc...) where the argument is a linear function of the variable, like . If you substitute in this case, you will simplify the integral and always end up with a factor of . Let's do a couple of examples to illustrate:

Example 1: -- This is a function of 2 x , which is of the form for a = 2 and b =0. Make the substitution u = 2 x . Then , or equivalently . If we do the substitution we get that = -- as promised, there is a factor of in the answer.

Example 2: -- This is a function of 3 x , so we're expecting a 1/3 in the answer. Make the substitution , Then or equivalently . Do the substitution and get = .

Example 3: One more: -- This is a function of 6 x - 3 -- so we expect a factor of 1/6. Make the substitution , so we get , or equivalently . Do the substitution and get = = + C = .

You try a couple: Calculate: , ,

So that you can check your understanding, I got , and as the answers.