Overview
Sometimes, a derivative is done using the chain rule, and it leaves with an equation that, at first glance, can look intractable when we are attempting to integrate it. In those cases, although the problem may look difficult or often will look like something that can only be solved through the use of Integration by Parts, there often exists a much simpler solution: that of U-Substitution.
Idea
Suppose we have an equation in the form ∫u’(x)f(u(x))dx. Then we can do a substitution in order to make the equation easier to integrate in a way that we will see shortly.
Although the way it was phrased above may make it sound intimidating, in reality the process is very straightforward.
Example
Suppose we wish to find ∫x3sin(x4)dx Although this may look tricky, once we notice that the derivative of x4 includes x3, it becomes much more straightforward.
We let u=x4
Then in that case, du/dx=4x3, or du/4 = x3dx
If we move things around in the initial integral, we can note that
∫x3sin(x4)dxdx = ∫sin(x4)x3dx
Now, since x4 = u, and x3dx = du/4, we can substitute in for the x as such
∫sin(u)du/4 = 1/4 ∫sin(u)du
Which is ¼(-cos(u))+C = -¼cos(u)+C
And now we need to substitute back for u, so
-¼cos(u)+C = -¼cos(x4)+C
Common Mistakes
If all of the xs have not been eliminated, then we cannot go through with the substitution. A good choice of u will leave no x or dx in the equation at all, only u and du.
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