U-Substitution
Remember the chain rule for differentiation?
ddxF(g(x))=F′(g(x))g′(x)
U-substitution is the chain rule done in reverse. Suppose that
F′=f.
∫f(g(x))g′(x)dx=F(g(x))+C
Now let's substitute
u=g(x).
∫f(g(x))g′(x)dx=F(u)+C
But we also know that the following is true.
∫f(u)du=F(u)+C
Therefore, it should be obvious that
∫f(g(x))g′(x)dx=∫f(u)du
This means that
du=g′(x)dx. That's right, we can treat
du as a differential. So how does this help us integrate? We can pick an
u inside the integrand and find its
du. Then we substitute the variable
x out completely, so our integrand is a function of
u and we are integrating with respect to
u. Once we finish evaluating the integral, we replace
u with whatever we picked when we started the integration.
Let's try a simple example. Suppose we have the following integral.
∫cos(2x+3)dx
Let's pick
u=2x+3. Then
du=2dx. So
dx=12du. Making these substitutions
∫cos(2x+3)dx=∫cos(u)12du
But
12 is a constant and may be pulled out of the integral. So
∫cos(u)12du=12∫cos(u)du
Integrating with regard to
u, we have
12∫cos(u)du=12sin(u)+C
Finally, replace
u with
2x+3, which is what we chose earlier.
12sin(u)+C=12sin(2x+3)+C
And we are done. Pretty cool, right?
Next week, we start Chapter 6, Applications of Integration. And I am taking my first midterm this week, as well. :'(