Calculus Section 4.5 Integration by Substitution
-Use a change of variables to find an indefinite integral
Homework: page 304 #’s 7, 8, 13, 14, 19, 20,
27, 28, 43-47, 49, 51
All of the antiderivatives that we’ve found so far have been simple. We have found them by either using the
integral’s “power rule” or recognizing the inverse of a trig derivative. Now we will learn a technique that is
comparable to the _________________________ from derivatives called u-substitution.
Antidifferentiation of a Composite Function
Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is
differentiable on its domain and F is an antiderivative of f on I, then
 f ( g ( x)) g '( x)dx  F ( g ( x))  C
Or, if you allow u = g(x), then du = g’(x)dx and
 f (u )du  F (u )  C
The key to solving any u-substitution problem is to look for a u-function whose __________________ will
match up with other parts of the function. We’re trying to simplify each integrand into something that we’ve
seen before and already know how to take the integral of using the integral “power rule.”
Examples)
1)  2 x( x 2  1) 4 dx
2)  3x 2 x3  1dx
3)  sec2 x(tan x  3)dx
4)  5cos(5 x)dx
More Examples)
5)
 x( x
2
 1) 2 dx
6)
x
2
x3  2dx
7)  2sec 2 x(tan x  3)dx
With this integration technique, it is only possible to move ____________________ out of the integral sign.