Tutorials on Advanced Math and Computer Science Concepts

# Substitution Rule

When evaluating integrals, it is often helpful to make substitutions to help see patterns in functions. We often apply this to functions that come in the form of a chain rule derivative.

The Substitution Rule: If u = g(x) is a differentiable function, then $\int f(g(x))g'(x)dx = \int f(u) du$

Example: Determine the integral of $\int 2x*\sqrt{1 + x^2}dx$

This function is in the form required for substitution rule. In this case, $g(x) = 1 + x^2$. If we allow $u = 1+x^2$, then $du = u' = 2x dx$

This tells us that $du = 2x dx$, so when we place the du in the equation, it can take the place of $2x dx$.

From here, we substitute in what we have currently found. $\int \sqrt{u} du$. The $\sqrt{1 + x^2}$ becomes $\sqrt{u}$ since we are letting $u = 1 + x^2$. The $2x dx$ becomes $du$, since $du = 2x dx$, allowing us to remove the 2x and replace it with du. From here, we can solve the integral like any other one we have seen so far.

$\int \sqrt{u} du = \frac{2}{3} u^{\frac{3}{2}} + C$

This gives us the answer in terms of u. We want to convert the u back into g(x), so we can replace every instance of u with g(x) to get:

$\frac{2}{3} (1 + x^2)^{\frac{3}{2}} + C$

Substitution rule allows us to easily see how to take an integral when the initial function looks more complex. The general steps you want to follow is to determine u and du, then make the proper substitutions in the integral. Sometimes, $du \ne g'(x) dx$, so we will need to add a constant or other component to make the substitution work.

Example: Determine the integral of $\int \sqrt{2x + 1} dx$

At first glance, this may not look like it fits the proper substitution rule form, however we can manipulate it to make it work. If we choose to let $u = 2x + 1$, it would mean that $du = 2 dx$. This means that $dx = \frac{1}{2} du$ From here, we can substitute in u to get:

$\int \sqrt{u} * \frac{1}{2} du = \frac{1}{3} * u^{\frac{3}{2}} + C = \frac{1}{3}*(2x+1)^{\frac{3}{2}} + C$

It's important to note that if you are evaluating a definite integral, you should swap u for g(x) before attempting to evaluate. You could also alternatively convert the x values to u values, depending on what feels more comfortable.

Example: Find $\int_{0}^{2} \sqrt{2x+1} dx$

We found that $\int \sqrt{2x+1} dx = \frac{1}{3} * u^{\frac{3}{2}} + C$

To evaluate the integral from x = 0 to x = 2, we have two choices. First, we can convert the u back to g(x) and input the x values.

$\frac{1}{3} * u^{\frac{3}{2}} + C = \frac{1}{3}*(2x+1)^{\frac{3}{2}} + C$
$=\frac{1}{3}*(4+1)^{\frac{3}{2}} - \frac{1}{3} * 1$
$=3.39345$

We could also convert the x values into equivalent u values, and solve while the integral is still in the u form.

$u(x) = 2x+1$
$u(2) = 5$
$u(0) = 1$

$\frac{1}{3}* u^{\frac{3}{2}} = \frac{1}{3}*5^{\frac{3}{2}} - \frac{1}{3} = 3.39345$

Both methods give you the exact same solution, so it really is a matter of personal preference when choosing which you want to use. In many cases, finding an equivalent u value and putting it in the integral gives an easier expression compared to substituting back in g(x).