## 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(x ^{2}) 2x dx = sin(x^{2}) + 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(x^{2}) 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(x^{2}) 6x dx = 3∫cos(x^{2}) 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=x ^{2}** back again:

3 sin(x^{2}) + C

Done!

Now let’s try a slightly harder example:

Example: ∫x/(x^{2}+1) dx

Let me see … the derivative of x^{2}+1 is 2x … so how about we rearrange it like this:

∫x/(x^{2}+1) dx = ½∫2x/(x^{2}+1) dx

Then we have:

Then integrate:

½∫1/u du = ½ ln**|**u**|** + C

Now put **u=x ^{2}+1** back again:

½ ln(x^{2}+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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