U Substitution Formula

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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 :

$\int f (g (x)) g^{'} (x) d x = \int f (u) d u$

where,

u = g(x)
du = $g^{'} (x) d x$

Let us see how to use the u substitution formula in the following solved examples section.

Examples Using U Substitution Formula

Example 1: Integrate $\int (2 x + 6) (x^{2} + 6 x)^{6} d x$

using u substitution formula.

Solution:

Let u = $x^{2} + 6 x$

So that, du = (2x+6)dx.

Substitute the value of u and du in $\int (2 x + 6) (x^{2} + 6 x)^{6} d x$

, replacing all forms of x, getting

Using U Substitution Formula,

$\int (2 x + 6) (x^{2} + 6 x)^{6} d x = \int (x^{2} + 6 x)^{6} (2 x + 6) d x$

Example 2: Integrate $\int (2 - x)^{8} d x$

Solution:

Let u = (2 – x)

So that, du = (-1)dx.

Substitute the value of u and du in $\int (2 - x)^{8} d x$

, 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.