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).

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