This cheat sheet covers how to use -substitution to evaluate integrals by reversing the chain rule. Students need it because many integrals look complicated until the inside function and its derivative are matched. The examples help students choose , rewrite , change bounds when needed, and avoid common algebra errors.
It is designed as a formula-forward reference for grades 11-12 calculus practice.
The core idea is to set when an integral contains a composite function such as . Then compute and rewrite the whole integral in terms of . For indefinite integrals, integrate in and substitute back to .
For definite integrals, either change the bounds to -values or substitute back before applying the original -bounds.
Key Facts
- The basic -substitution pattern is when and .
- For an indefinite integral, after finding , substitute back into the answer and add .
- For a definite integral, if , then .
- A constant multiplier can be adjusted because may require multiplying and dividing by the same constant.
- For powers, the pattern works when .
- For logarithmic forms, .
- For exponential forms, .
- For trigonometric forms, examples include and .
Vocabulary
- -substitution
- A method for rewriting an integral using so the integral becomes simpler to evaluate.
- Composite function
- A function inside another function, such as where is the inside function.
- Differential
- The expression that tells how a small change in relates to a small change in .
- Antiderivative
- A function whose derivative is the integrand, so .
- Definite integral bounds
- The lower and upper limits of integration, which must be changed from -values to -values when using -substitution directly.
- Constant of integration
- The added to an indefinite integral because antiderivatives differ by a constant.
Common Mistakes to Avoid
- Choosing only the outside function for is wrong because should usually be the inside expression whose derivative also appears in the integrand.
- Forgetting to rewrite using is wrong because the integral must be completely converted into the new variable before integrating.
- Dropping a constant factor is wrong because if but the integral has , the substitution needs the factor .
- Using the original bounds after changing to is wrong because bounds like and must become and .
- Forgetting to substitute back in an indefinite integral is wrong because the final antiderivative must be written in terms of the original variable unless the problem says otherwise.
Practice Questions
- 1 Evaluate using -substitution.
- 2 Evaluate using -substitution.
- 3 Evaluate by changing the bounds to -values.
- 4 Explain why is a better substitution than for an integral containing .
Understanding U-Substitution Master Examples
The most useful skill is learning to spot a quantity that changes as one unit. In an expression such as the square root of three x squared plus one, the expression three x squared plus one behaves like a single input to the square root function. Its rate of change is six x.
When a nearby factor supplies six x, or a constant multiple of it, the integral has the right structure for substitution. This is why students should scan for repeated groups inside powers, roots, exponentials, logarithms, and trig functions.
Circle the repeated group first. Then compare its derivative with the factors outside it.
Differential matching is more than choosing a convenient letter. The replacement must remove every remaining x from the integral. For example, if the new variable is three x squared plus one, then its differential contains six x times dx.
If the original integral has only x times dx, a factor of one sixth is needed. That factor belongs outside the integral. Constants are often the only obstacle, so check them carefully before integrating.
A common mistake is to replace the repeated expression with u but leave an x elsewhere. That creates a mixed-variable integral, which cannot be evaluated using ordinary single-variable rules. Solving for the needed differential factor before rewriting helps prevent this error.
Definite integrals need extra care because the endpoints belong to the variable being used. Once the bounds are converted, both bounds must stay in the new variable through the rest of the calculation. Mixing an old x bound with a new u bound has no meaning.
The order of the bounds matters too. If the substitution decreases as x increases over the interval, the new lower bound can be larger than the new upper bound. The resulting negative sign is correct and should not be changed by hand.
Logarithmic answers need another check. The absolute value in the logarithm protects the result on intervals where the inner expression is negative, but the original integrand must still be defined throughout the interval.
Substitution appears whenever one quantity depends on another quantity. In physics, a changing position may be placed inside a velocity, force, or energy formula. In population models, a time-dependent rate can occur inside an exponential expression.
The calculus step is the same because the chain rule describes nested change in every setting. The best final check is differentiation. Differentiate the answer and see whether it returns the original integrand exactly, including constants and signs.
If it does not, inspect the derivative of the chosen inner expression first. Some integrals only resemble substitution problems. A product with no matching derivative, or a sum of unrelated functions, may need a different method such as integration by parts, trig identities, or algebraic rewriting.