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.

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

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).