U Substitution Formula
U-substitution is also known as integration by substitution in calculus, u-substitution formula is a method for finding integrals. The fundamental theorem of calculus generally used for finding an antiderivative. Due to this reason, integration by substitution is an important method in mathematics. The u-substitution formula is another method for the chain rule of differentiation. This u substitution formula is similarly related to the chain rule for differentiation. In the u-substitution formula, the given function is replaced by ‘u’ and then u is integrated according to the fundamental integration formula. After integration, we resubstitute the actual function in place of u. Let us learn more about the u-substitution formula in the upcoming sections.
What Is U Substitution Formula?
In the U substitution formula, the main function is replaced by ‘u’ and then the variable u is integrated according to the fundamental integration formula but after integration we resubstitute the actual function in place of u. U substitution formula can be given as :
where,
- u = g(x)
- du =
Let us see how to use the u substitution formula in the following solved examples section.
Examples Using U Substitution Formula
Example 1: Integrate
using u substitution formula.
Solution:
Let u =
So that, du = (2x+6)dx.
Substitute the value of u and du in
, replacing all forms of x, getting
Using U Substitution Formula,
Example 2: Integrate
Solution:
Let u = (2 – x)
So that, du = (-1)dx.
Substitute the value of u and du in
, replacing all forms of x, getting
Using U substitution formula,
Integration by Substitution
“Integration by Substitution” (also called “u-Substitution” or “The Reverse Chain Rule”) is a method to find an integral, but only when it can be set up in a special way.
The first and most vital step is to be able to write our integral in this form:
Then we can integrate f(u), and finish by putting g(x) back as u.
Like this:
So ∫cos(x2) 2x dx = sin(x2) + C
That worked out really nicely! (Well, I knew it would.)
But this method only works on some integrals of course, and it may need rearranging:
Example: ∫cos(x2) 6x dx
Oh no! It is 6x, not 2x like before. Our perfect setup is gone.
Never fear! Just rearrange the integral like this:
∫cos(x2) 6x dx = 3∫cos(x2) 2x dx
(We can pull constant multipliers outside the integration, see Rules of Integration.)
Then go ahead as before:
3∫cos(u) du = 3 sin(u) + C
Now put u=x2 back again:
3 sin(x2) + C
Done!
Now let’s try a slightly harder example:
Example: ∫x/(x2+1) dx
Let me see … the derivative of x2+1 is 2x … so how about we rearrange it like this:
∫x/(x2+1) dx = ½∫2x/(x2+1) dx
Then we have:
Then integrate:
½∫1/u du = ½ ln|u| + C
Now put u=x2+1 back again:
½ ln(x2+1) + C
And how about this one:
Example: ∫(x+1)3 dx
Let me see … the derivative of x+1 is … well it is simply 1.
So we can have this:
We can take that idea further like this:
Example: ∫(5x+2)7 dx
If it was in THIS form we could do it:
Now get some practice, OK?
Practice Problems – U Substitution for Integration
In the following practice problems, students will tackle increasingly difficult integration problems using u-substitution. By the end of the problems, students will have more confidence in choosing u for the technique and be able to understand the process of u-substitution.
Practice Problems
On the following problems, use u-substitution to integrate. Clearly indicate your choice for u and the work needed to rewrite the integral in terms of u and du.
Solutions
1. Choosing u to be the expression in the parentheses is often the best choice, so we will choose:
5. Since we don’t have a simple formula for the anti derivative of a natural logarithm, we will choose:
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