Integration Methods to Know for AP Calculus AB/BC (AP)

What You Need to Know

Integration on AP Calc AB/BC is mostly about recognizing patterns and picking the right method fast. You’re not expected to invent clever antiderivatives from scratch—you’re expected to:

• Use the Fundamental Theorem of Calculus to evaluate definite integrals.
• Find antiderivatives with a small toolbox of techniques.
• Handle definite integrals, improper integrals, and numerical integration correctly.

Core idea: an antiderivative of $f$ is a function $F$ such that $F'(x)=f(x)$, and

• FTC Part 1 (accumulation): If $g(x)=\int_a^x f(t)\,dt$, then $g'(x)=f(x)$.
• FTC Part 2 (evaluation): $\int_a^b f(x)\,dx = F(b)-F(a)$ where $F'(x)=f(x)$.

When you choose a method:

• If you see “inside function derivative nearby” → use $u$-substitution.
• If you see a product like polynomial $\times$ trig/exp/log → try integration by parts.
• If you see a rational function $\frac{P(x)}{Q(x)}$ → do long division and/or partial fractions.
• If you see trig powers/products → use trig identities.
• If it’s not integrable nicely (or asked explicitly) → use Trapezoidal/Simpson’s Rule.

Critical reminder: On FRQs, method + correct setup is often worth a lot even if algebra gets messy.

Step-by-Step Breakdown

1) $u$-Substitution (Reverse Chain Rule)

Use when the integrand looks like $f(g(x))g'(x)$.

1. Choose $u=g(x)$ (the “inside”).
2. Compute $du=g'(x)\,dx$.
3. Rewrite the integral fully in $u$.
4. Integrate in $u$.
5. Substitute back $u=g(x)$.
6. If definite, either:
   • change bounds: $u(a)=g(a)$ and $u(b)=g(b)$ (cleanest), or
   • switch back to $x$ and then plug in.

Mini-example (indefinite):

• $\int 2x\cos(x^2)\,dx$
• Let $u=x^2$, then $du=2x\,dx$.
• Integral becomes $\int \cos(u)\,du = \sin(u)+C = \sin(x^2)+C$.

2) Integration by Parts

Use for products where substitution doesn’t cleanly work (especially involving $\ln x$, inverse trig, polynomials with $e^x$ or trig).

Formula:
$\int u\,dv = uv - \int v\,du$

Steps:

1. Pick $u$ (differentiate it) and $dv$ (integrate it).
2. Compute $du$ and $v$.
3. Plug into $\int u\,dv = uv - \int v\,du$.
4. Simplify; sometimes you’ll need to do it twice (common with trig $\times$ trig).
5. If the original integral reappears, solve algebraically.

Mini-example:

• $\int x e^x\,dx$
• Choose $u=x$, $dv=e^x\,dx$. Then $du=dx$, $v=e^x$.
• $\int x e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C = e^x(x-1)+C$.

3) Rational Functions: Long Division + Partial Fractions (BC-heavy)

When integrating $\frac{P(x)}{Q(x)}$.

Step A: Long division first if needed

1. If $\deg(P)\ge \deg(Q)$, do polynomial long division:
   $\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}$
2. Integrate $S(x)$ easily, then focus on proper fraction $\frac{R(x)}{Q(x)}$.

Step B: Partial fractions (for proper rational functions)

1. Factor the denominator $Q(x)$ completely (over reals).
2. Set up the decomposition based on factor types:
   • Distinct linear: $\frac{A}{x-a}$
   • Repeated linear: $\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots$
   • Irreducible quadratic: $\frac{Ax+B}{x^2+px+q}$
   • Repeated irreducible quadratic: stack powers similarly.
3. Solve for constants (plug values or match coefficients).
4. Integrate term-by-term.

Mini-example (distinct linears):

• $\int \frac{1}{x^2-1}\,dx = \int \frac{1}{(x-1)(x+1)}\,dx$
• $\frac{1}{(x-1)(x+1)}=\frac{A}{x-1}+\frac{B}{x+1}$
• Solve: $1=A(x+1)+B(x-1)$ gives $A=\frac{1}{2}$, $B=-\frac{1}{2}$.
• Integral: $\frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right|+C$.

4) Trig Integrals via Identities (not “trig substitution”)

Common AP patterns:

A) Powers of $\sin x$ and $\cos x$

1. If one power is odd, peel off one factor and convert the rest using:
   $\sin^2 x = 1-\cos^2 x \quad \text{or} \quad \cos^2 x = 1-\sin^2 x$
2. If both are even, use half-angle identities:
   $\sin^2 x = \frac{1-\cos(2x)}{2},\quad \cos^2 x = \frac{1+\cos(2x)}{2}$

B) Powers of $\tan x$ and $\sec x$

• If $\sec^2 x$ is present, let $u=\tan x$.
• If $\sec x\tan x$ is present, let $u=\sec x$.
• Use identities:
  $1+\tan^2 x = \sec^2 x,\quad \sec^2 x-1=\tan^2 x$

5) Improper Integrals (limits)

You have an improper integral if:

• infinite limit: $\int_a^{\infty} f(x)\,dx$ or $\int_{-\infty}^b f(x)\,dx$
• vertical asymptote/discontinuity in interval, e.g. $\int_a^b \frac{1}{x-c}\,dx$ with $c\in[a,b]$

Steps:

1. Rewrite as a limit:
   $\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx$
2. Evaluate the antiderivative and take the limit.
3. If the limit is finite → converges; otherwise diverges.

6) Numerical Integration (Calculator-friendly)

If you’re given a table of values or asked to approximate:

Trapezoidal Rule (works on any partition):

1. Partition $[a,b]$ into $n$ equal subintervals of width $\Delta x = \frac{b-a}{n}$.
2. Apply:
   $T_n = \frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n-1})+f(x_n)\right]$

Simpson’s Rule (requires even $n$):

1. Use even number of subintervals $n$.
2. Apply:
   $S_n = \frac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right]$

If the problem gives unequal subintervals, Simpson’s Rule usually does not apply; Trapezoidal still can.

Key Formulas, Rules & Facts

Integration method “trigger table”

Handy trig/identity list (high-yield)

• Pythagorean:
  $\sin^2 x+\cos^2 x=1$
  $1+\tan^2 x=\sec^2 x$
  $1+\cot^2 x=\csc^2 x$
• Half-angle:
  $\sin^2 x=\frac{1-\cos(2x)}{2},\quad \cos^2 x=\frac{1+\cos(2x)}{2}$
• Basic trig antiderivatives:
  $\int \cos x\,dx=\sin x + C$
  $\int \sin x\,dx=-\cos x + C$
  $\int \sec^2 x\,dx=\tan x + C$
  $\int \csc^2 x\,dx=-\cot x + C$
  $\int \sec x\tan x\,dx=\sec x + C$
  $\int \csc x\cot x\,dx=-\csc x + C$

Definite integral properties you actually use

• Linearity:
  $\int_a^b (cf+g)=c\int_a^b f + \int_a^b g$
• Reversal:
  $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$
• Additivity:
  $\int_a^b f = \int_a^c f + \int_c^b f$
• Symmetry on $[-a,a]$:
  • If $f$ is even: $\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx$
  • If $f$ is odd: $\int_{-a}^a f(x)\,dx = 0$

Examples & Applications

Example 1: Definite integral with $u$-sub (change bounds)

Compute $\int_0^1 6x(3x^2+2)^4\,dx$.

• Let $u=3x^2+2$, so $du=6x\,dx$.
• Change bounds: when $x=0$, $u=2$; when $x=1$, $u=5$.
• Then $\int_0^1 6x(3x^2+2)^4\,dx = \int_2^5 u^4\,du = \left.\frac{u^5}{5}\right|_2^5 = \frac{5^5-2^5}{5}$.

Key exam insight: changing bounds avoids back-sub and is often cleaner.

Example 2: By parts with $\ln x$ (classic)

Compute $\int \ln x\,dx$.

• Write $\int \ln x\,dx = \int 1\cdot \ln x\,dx$.
• Choose $u=\ln x$, $dv=dx$. Then $du=\frac{1}{x}dx$, $v=x$.
• $\int \ln x\,dx = x\ln x - \int x\cdot \frac{1}{x}\,dx = x\ln x - \int 1\,dx = x\ln x - x + C$.

Key exam insight: inverse trig and logs almost always scream “parts.”

Example 3: Partial fractions with repeated factor

Compute $\int \frac{1}{x(x-1)^2}\,dx$.

• Setup:
  $\frac{1}{x(x-1)^2}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{(x-1)^2}$
• Multiply through by $x(x-1)^2$:
  $1=A(x-1)^2+Bx(x-1)+Cx$
• Plug convenient values:
  • $x=0$: $1=A(1)\Rightarrow A=1$
  • $x=1$: $1=C\Rightarrow C=1$
• Use another value (say $x=2$):
  $1=1\cdot 1^2 + B\cdot 2\cdot 1 + 1\cdot 2 \Rightarrow 1=1+2B+2 \Rightarrow B=-1$
• Integrate:
  $\int \left(\frac{1}{x}-\frac{1}{x-1}+\frac{1}{(x-1)^2}\right)dx = \ln|x| - \ln|x-1| - \frac{1}{x-1}+C$

Key exam insight: repeated factors require a whole stack of terms.

Example 4: Trig power integral

Compute $\int \sin^3 x\,dx$.

• Peel off one $\sin x$:
  $\int \sin^3 x\,dx = \int \sin^2 x\sin x\,dx = \int (1-\cos^2 x)\sin x\,dx$
• Let $u=\cos x$, $du=-\sin x\,dx$.
• Integral becomes:
  $-\int (1-u^2)\,du = -\left(u-\frac{u^3}{3}\right)+C = -\cos x + \frac{\cos^3 x}{3}+C$

Key exam insight: odd power of $\sin$ or $\cos$ usually means “save one, convert the rest.”

Common Mistakes & Traps

1. Forgetting $du$ (or not matching it)

   • Wrong: letting $u=g(x)$ but not converting $dx$ terms correctly.
   • Fix: after substitution, the integral must be entirely in $u$ (including $du$).

2. Not changing bounds on definite $u$-sub (then mixing variables)

   • Wrong: switching to $u$ but still plugging in $x=a,b$.
   • Fix: either change bounds to $u(a),u(b)$ or revert back to $x$ before evaluating.

3. Dropping absolute values in log answers

   • Wrong: writing $\ln(x)$ instead of $\ln|x|$.
   • Fix: for indefinite integrals, default to $\ln|\cdot|$ unless domain is explicitly restricted.

4. Using integration by parts when $u$-sub is simpler (or vice versa)

   • Wrong: forcing parts on something like $\int 2x\cos(x^2)dx$.
   • Fix: always check for inside-derivative structure first.

5. Skipping long division before partial fractions

   • Wrong: trying PF when $\deg(P)\ge\deg(Q)$.
   • Fix: divide first; PF only applies cleanly to proper rational functions.

6. Incorrect partial fraction form (especially repeats/quadratics)

   • Wrong: for $\frac{1}{(x-1)^2}$ writing only $\frac{A}{x-1}$.
   • Fix: repeated linear factors require $\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots$; irreducible quadratics need $Ax+B$ on top.

7. Simpson’s Rule with odd $n$ or unequal spacing

   • Wrong: applying Simpson when $n$ is odd or the table step size changes.
   • Fix: Simpson requires equal spacing and even $n$. Otherwise use trapezoids.

8. Improper integrals: evaluating at the asymptote instead of using limits

   • Wrong: plugging in the discontinuity like it’s a normal endpoint.
   • Fix: rewrite with a limit; you must state convergence/divergence.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can apply FTC: find $F$ then compute $F(b)-F(a)$.
• You can spot and execute $u$-sub, including changing bounds for definite integrals.
• You know integration by parts and can pick $u$ using LIATE.
• You can integrate rational functions: long division first, then partial fractions.
• You can set up partial fractions for distinct/repeated linear and irreducible quadratic factors.
• You can handle trig power integrals using identities (odd/even strategy).
• You can evaluate improper integrals using limits and state converge/diverge.
• You can approximate with Trapezoidal and Simpson’s Rule (and know Simpson needs even $n$).

You’ve got enough tools—your job is just picking the right one quickly and executing cleanly.