Method: U-Substitution

As you know from earlier antiderivative work Definite Integrals require an extra step at the end of the method compared to Indefinite Integrals. The extra step is the final step when you use the bounds to evaluate the integral to get a definite value for the Net area between the curve and the x-axis .

When applying the u-substitution method the Definite Integral versions will require an additional added step, and again it will be directly related to the bounds of a Definite Integral. Otherwise, the method for finding the antiderivative using u-substitution will be the exact same between Definite and Indefinite Integrals.

Keep in mind that the main purpose of the u-substitution method is to take the integral you have been given and can’t solve , and turn it into an integral that you are able to solve . You will do what I call side work, work off to the side of your main integral problem, that you will substitute back into the main integral. When doing u-substitution problems I really do split my paper down the middle. On the left side I track the main integral that I am actually trying to solve, and on the right side I do the side work that I will substitute back into the main integral.

Step 1: Simplify and look for algebraic rewrites.

This is your new version of Option 1 from limits. This means it is the first thing you will always want to do before you start actually applying a derivative rule. You must ensure that the equation is ready to have a derivative taken.

• Roots or Radicals
• x’s on the bottom of a fraction.
• Trig functions raised to a power.

Step 2 (Beginning of the side work) : Choose your u .

I have developed my own priority list to help me choose my u . I work my way down the priority list, one option at a time until I find what I have going on in my specific integral.

The U Priority List

1)      Priority 1: $\ln \left(x\right)$

If you see a $\mathit{ln}\left(x\right)$, that is always going to be the first option you will want to choose for your u .

2)      Priority 2: The Inside piece

When I say inside piece, I mean what you would have considered the inside piece when you were doing a chain rule derivative. The piece that is INSIDE a set of parenthesis or root ($f\left(x\right)= \left({3x}^2+10x\right){\left(x^3+{5x}^2\right)}^3$ or $f\left(x\right)= \frac{1}{6}\sqrt{6x+9}$), or INSIDE a trig function ($\mathit{cos}\left(x^3\right)$), or exponential exponent ($e^{\left({5x}^2\right)}$).

3)      Priority 3: The Bottom of a Fraction

$f\left(x\right)=\frac{1}{15x–19}$

4)      General Priority: The piece with the highest power of x .

You almost always want the piece that has the highest power of x .

Write down your u on your paper to the right of your main integral.

Step 3 (Side work) : Take the derivative of your u , and always use differential notation to represent the derivative (i.e. $\frac{\mathit{dy}}{\mathit{dx}}$, not $f\prime \left(x\right)$).

Step 4 (Side work) : Solve the derivative you just created for $\mathit{dx}$.

This means you want to treat $\mathit{dx}$ like any other variable and get it alone one side of your equal’s sign.

Step 5 (Side work for Definite Integrals only ): Adjust the original bounds, $\underset{a}{\overset{b}{\int }}$ of your main integral.

Keep in mind that the original bounds on your integral are bounds in terms of your original variable . If your main integral is talking about x ’s, those bounds are x-values. Once you perform your actual u-substitution, your new integral will be talking in the language of u ’s, which means you need your bounds to be u-values.

To adjust your bounds , you will take the u – equation you selected in Step 2 , and plug your original bounds, $\underset{a}{\overset{b}{\int }}$, into the u – equation for the x-values to get your new bounds , u-values.

Step 6: Substitute back into your main integral.

You work your way through your main integral left to right, and each piece in the main integral is either one of the pieces you are substituting from your side work or it stays.

• Substitute in u for the piece that you chose as your u in Step 1 .
• Substitute in for the $\mathit{dx}$what you found in Step 4 .
• (Definite Integrals only) Substitute in your new upper and lower bounds that you found in Step 5 .

Step 7: Simplify the updated main integral you just substituted into.

If you have done everything properly you should be left with an equation that talks only the language of u .

Any x ’s that were left in the main integral better cancel out with x ’s that you substituted in or you probably messed up somewhere. You either chose the wrong u , or got some portion of your side work algebra wrong. You should stop and relook at your previous work.

In some rare cases (often times a test given by your teacher and not as often on the actual AP Calculus exam) you might have to:

• Perform a 2 ^nd u-substitution process.
• Solve your u -equation from Step 2 for x (i.e., get x alone on one side of the equals), and substitute that value back into the main integral for any x-values that did not cancel out.

One important simplification step, is to rewrite your terms in what is “standard” form, numbers first and then variables.

You should also take this opportunity to slide any constants up to the front of the equation. The constant will often times be a fraction. You are more comfortable seeing equations looking like standard form. Therefore, you are better able to identify the antiderivative method you will need to apply.

Step 8: Treat the simplified main integral from Step 7 like a brand-new math problem , and begin your standard antiderivative methods.

I always tell my students this your “reward” for all your hard work to get to the is point, a brand-new math problem .

If you have done everything properly, you should now be looking at an integral that you actually do know how to do based on your standard antiderivative rules (i.e., power rule or special case).

Step 9 (Definite Integrals only ) : Evaluate the antiderivative at the bounds of your integral.

You need to finish evaluating like you would on any Definite Integral problem using the Top – Bottom method. Plug the upper bound into your antiderivative and subtract off the lower bound plugged into your antiderivative.

Step 9 (Indefinite Integrals only ) : Substitute back in for the u ’s in the antiderivative result.

I always say, “You don’t care about u , you care about what u was.”

The main integral you were being asked to find all the way back at the beginning of this process did not have u ’s as the variables. You need to substitute back in what you chose for u in Step 2 for every u in your result from Step 8 .