Integration by substitution (u-sub) — AP Calculus AB
1. What Is Integration by Substitution (u-sub)? ★☆☆☆☆ ⏱ 3 min
Integration by substitution (commonly shortened to u-sub) is the core advanced integration technique for AP Calculus AB, designed explicitly to reverse the chain rule from differentiation. Per the AP CED, this topic accounts for 10-15% of Unit 6 weight, and you can expect 2-3 MCQ questions and at least one FRQ part requiring u-sub on every full AP exam.
The method works by rewriting a complicated integral of a composite function in terms of a new variable $u$, which is chosen to be the inner function of the composite. This turns an unfamiliar integral into a basic integral you already know how to solve.
2. U-Substitution for Indefinite Integrals ★★☆☆☆ ⏱ 4 min
U-sub reverses the chain rule relationship. For a composite function $f(x) = g(h(x))$, the chain rule gives:
f'(x) = g'(h(x)) \cdot h'(x)
For an integral of the form $\int g'(h(x)) \cdot h'(x) dx$, set $u = h(x)$, so $du = h'(x) dx$. Substituting gives:
\int g'(u) du = g(u) + C = g(h(x)) + C
If you are only missing a constant coefficient, you can adjust by rearranging the differential and factoring the reciprocal constant out of the integral. Non-constant adjustments are never required on AP Calculus AB.
Exam tip: Always substitute back to the original variable $x$ for indefinite integrals. AP exam graders will deduct full points for a correct antiderivative left in terms of $u$.
3. U-Substitution for Definite Integrals (Changing Bounds) ★★☆☆☆ ⏱ 4 min
For definite integrals, you can either substitute back to $x$ after integrating, or change the bounds of integration to match $u$, which eliminates back-substitution entirely. The bounds-changing method is faster and less error-prone on the AP exam, so it is the recommended approach.
When changing bounds for $\int_{x=a}^{x=b} f(h(x)) h'(x) dx$, after setting $u = h(x)$, calculate the lower $u$-bound as $u = h(a)$ and the upper $u$-bound as $u = h(b)$. The integral becomes:
\int_{h(a)}^{h(b)} f(u) du
You integrate directly with respect to $u$ and evaluate, with no back-substitution needed. This method is especially common for AP MCQ where you only need the final numerical value.
Exam tip: Write down your new u-bounds immediately after setting $u$, before you rewrite the integral. This eliminates the common mistake of accidentally using the original x-bounds when integrating with respect to $u$.
4. U-Choice Strategy for Non-Linear Inner Functions ★★★☆☆ ⏱ 3 min
Most u-sub problems on the AP exam use non-linear inner functions, so having a consistent strategy for choosing $u$ is critical. The number one rule of thumb for AP AB: if you see a function and its derivative (up to a constant multiple) in the integrand, the function is your $u$.
Common non-linear inner functions tested on AP AB include powers of trigonometric functions, logarithms, polynomials under roots, and exponential functions. If you end up needing a non-constant term of $x$ to complete $du$, you have almost certainly chosen the wrong $u$.
Exam tip: Never change your $u$ to adjust for a missing constant factor. Just rearrange the differential to get the correct multiple of $du$, and factor the constant out of the integral. Changing $u$ for a constant will always introduce unnecessary errors.
Common Pitfalls
Why: Students get used to the bounds-changing method for definite integrals and forget that indefinite integrals require an answer in the original variable
Why: Students rush and skip the step of calculating new bounds, or forget that the variable of integration changed
Why: Students memorize "pick the complicated part" but misidentify which part is the inner composite
Why: Students mix up algebra when rearranging the differential equation
Why: Students rush and add $C$ too early, incorrectly treating it as a variable