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.