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 which technique fits the integrand and executing cleanly. You’re expected to:

• Find antiderivatives (indefinite integrals) and evaluate definite integrals.
• Use the Fundamental Theorem of Calculus (FTC) to connect integrals and derivatives.
• Choose among a small set of core integration methods:
  • Basic antiderivative rules (power/exponential/trig)
  • $u$-substitution (reverse chain rule)
  • Integration by parts (BC emphasis; AB may appear conceptually)
  • Partial fractions (BC)
  • Trig identities / trig integrals (BC)
  • Improper integrals (BC)
  • Numerical integration: Trapezoidal Rule (AB/BC) and Simpson’s Rule (BC)

Core idea: Integration is pattern matching + algebra. If you can rewrite the integrand into a standard form, you’re 90% done.

Big exam reminder: Always check your limits of integration if you substitute, and always add $+C$ for indefinite integrals.

Step-by-Step Breakdown

1) Basic Antiderivative Rules + FTC

Use when the integrand is already a standard derivative pattern.

1. Rewrite the integrand (simplify, split sums).
2. Apply known antiderivatives term-by-term.
3. For definite integrals, evaluate with FTC:
   • If $F'(x)=f(x)$, then $\int_a^b f(x)\,dx = F(b)-F(a)$.

Mini-example:

• $\int (3x^2 - 4\cos x)\,dx = x^3 - 4\sin x + C$.
• $\int_0^{\pi} \sin x\,dx = [-\cos x]_0^{\pi} = (-\cos \pi)-(-\cos 0)=2$.

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

Use when you see a composite function and (up to a constant) its derivative.

1. Choose $u = \text{“inside”}$.
2. Compute $du = u'(x)\,dx$.
3. Rewrite the integral entirely in $u$ and $du$.
4. Integrate in $u$.
5. Convert back to $x$ (unless you changed definite bounds).
6. For definite integrals: either convert bounds to $u$-bounds _or_ switch back to $x$ before evaluating.

Mini-example:

• $\int 2x\,(x^2+5)^7\,dx$
  • Let $u=x^2+5$, then $du=2x\,dx$.
  • Integral becomes $\int u^7\,du = \frac{u^8}{8}+C = \frac{(x^2+5)^8}{8}+C$.

Decision point: If you can spot an “inside” function whose derivative is present (or can be created with a constant factor), use $u$-sub.

3) Integration by Parts (IBP) (Mostly BC)

Use for products like $x e^x$, $x\sin x$, $\ln x$, inverse trig, etc.

Key formula:

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

Steps:

1. Choose $u$ (differentiate it) and $dv$ (integrate it).
2. Compute $du$ and $v$.
3. Plug into $\int u\,dv = uv - \int v\,du$.
4. Repeat if the remaining integral still needs IBP.

Mini-example:

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

Decision point: If it’s a product and $u$-sub doesn’t fit, think IBP.

4) Partial Fractions (BC)

Use for rational functions $\frac{P(x)}{Q(x)}$ where $\deg P < \deg Q$ (after long division if needed).

1. If $\deg P \ge \deg Q$: do polynomial long division first.
2. Factor the denominator completely (over reals).
3. Set up partial fraction form:
   • Distinct linear: $\frac{A}{x-a}$
   • Repeated linear: $\frac{A}{x-a}+\frac{B}{(x-a)^2}+\cdots$
   • Irreducible quadratic: $\frac{Ax+B}{x^2+px+q}$
4. Solve for constants.
5. Integrate term-by-term (often logs + arctan).

Mini-example:

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

5) Trig Integrals / Trig Identity Rewrites (BC)

Use when powers of $\sin x$ and $\cos x$ appear.

1. If one power is odd: save one factor, convert the rest using $\sin^2 x = 1-\cos^2 x$ or $\cos^2 x = 1-\sin^2 x$, then $u$-sub.
2. If both are even: use half-angle identities:
   • $\sin^2 x = \frac{1-\cos(2x)}{2}$
   • $\cos^2 x = \frac{1+\cos(2x)}{2}$

Mini-example (odd power):

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

6) Improper Integrals (BC)

Use when bounds are infinite or the integrand blows up.

1. Replace the problematic bound or point with a variable.
2. Evaluate as a limit.
3. If the limit is finite: converges. If infinite/DNE: diverges.

Forms:

• Infinite interval: $\int_a^{\infty} f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx$
• Vertical asymptote at $a$: $\int_a^b f(x)\,dx = \lim_{t\to a^+}\int_t^b f(x)\,dx$ (or split if asymptote is inside).

Mini-example:

• $\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\frac{1}{b}+1\right)=1$ (convergent)

7) Numerical Integration (Trapezoidal + Simpson)

Use when you have a table of values or no elementary antiderivative.

Trapezoidal Rule (AB/BC):

1. Compute $\Delta x = \frac{b-a}{n}$.
2. Apply:

$T_n = \frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right]$

Simpson’s Rule (BC): requires even $n$.

1. Compute $\Delta x = \frac{b-a}{n}$.
2. Apply:

$S_n = \frac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right]$

Decision point: Simpson’s is usually more accurate (for smooth functions), but you must have even $n$.

Key Formulas, Rules & Facts

Must-know antiderivatives (AB/BC)

Fundamental Theorem of Calculus (FTC)

Quick “which method?” table

Trig identities you actually use for integrals

• $\sin^2 x + \cos^2 x = 1$
• $1+\tan^2 x = \sec^2 x$
• $\sin^2 x = \frac{1-\cos(2x)}{2}$
• $\cos^2 x = \frac{1+\cos(2x)}{2}$

Examples & Applications

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

Evaluate $\int_0^1 2x\sqrt{x^2+1}\,dx$.

• Let $u=x^2+1$, so $du=2x\,dx$.
• Change bounds: when $x=0$, $u=1$; when $x=1$, $u=2$.
• Integral becomes $\int_1^2 u^{1/2}\,du = \left[\frac{2}{3}u^{3/2}\right]_1^2 = \frac{2}{3}\left(2^{3/2}-1\right)$.

Exam angle: This tests whether you remember to adjust bounds (or correctly switch back).

Example 2: IBP classic

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

• Use IBP with $u=\ln x$ and $dv=dx$.
• Then $du=\frac{1}{x}dx$, $v=x$.
• $\int \ln x\,dx = x\ln x - \int x\cdot \frac{1}{x}\,dx = x\ln x - \int 1\,dx = x\ln x - x + C$.

Exam angle: A “no obvious antiderivative” problem that becomes easy with IBP.

Example 3: Partial fractions with logs

Evaluate $\int \frac{3x+5}{x^2+2x-3}\,dx$.

• Factor denominator: $x^2+2x-3=(x+3)(x-1)$.
• Set: $\frac{3x+5}{(x+3)(x-1)}=\frac{A}{x+3}+\frac{B}{x-1}$.
• Solve: $3x+5=A(x-1)+B(x+3)=(A+B)x+(-A+3B)$.
  • System: $A+B=3$ and $-A+3B=5$ gives $B=2$, $A=1$.
• Integrate: $\int \left(\frac{1}{x+3}+\frac{2}{x-1}\right)dx = \ln|x+3|+2\ln|x-1|+C$.

Exam angle: Factoring + algebra accuracy.

Example 4: Simpson’s Rule from a table (BC)

Approximate $\int_0^4 f(x)\,dx$ using Simpson’s Rule with $n=4$ and values $f(0)=1$, $f(1)=3$, $f(2)=2$, $f(3)=5$, $f(4)=4$.

• $\Delta x = \frac{4-0}{4}=1$.
• $S_4=\frac{1}{3}\left[f(0)+4f(1)+2f(2)+4f(3)+f(4)\right]$
• $S_4=\frac{1}{3}\left[1+4(3)+2(2)+4(5)+4\right]=\frac{1}{3}(1+12+4+20+4)=\frac{41}{3}$.

Exam angle: Coefficient pattern $1,4,2,4,\dots,2,4,1$ and even $n$.

Common Mistakes & Traps

1. Forgetting $+C$ (indefinite integrals)

   • Wrong: treating an antiderivative as a single function.
   • Fix: if there’s no bounds, add $+C$ automatically.

2. Bad $u$ choice or not converting everything to $u$

   • Wrong: after substituting, leaving stray $x$ terms.
   • Fix: solve for $x$ in terms of $u$ if needed, or choose a cleaner $u$.

3. Not changing bounds (or double-changing) in definite $u$-sub

   • Wrong: switching to $u$ but still plugging in $x=a,b$.
   • Fix: choose ONE approach:
     • Convert bounds to $u$ and stay in $u$, or
     • switch back to $x$ before evaluating.

4. Sign errors with trig derivatives/antiderivatives

   • Common slips: $\int \sin x\,dx$ and $\int \csc^2 x\,dx$.
   • Fix: memorize the “negative” ones: $\int \sin x\,dx=-\cos x + C$ and $\int \csc^2 x\,dx=-\cot x + C$.

5. IBP setup mistakes (mixing up $u$ and $dv$ or missing parentheses)

   • Wrong: using $\int u\,dv = uv + \int v\,du$ (sign wrong) or dropping a factor.
   • Fix: write the formula every time: $\int u\,dv = uv - \int v\,du$.

6. Partial fractions before long division

   • Wrong: trying PF when $\deg P \ge \deg Q$.
   • Fix: divide first, then decompose the proper fraction.

7. Dropping absolute values in log answers

   • Wrong: $\int \frac{1}{x-2}\,dx = \ln(x-2)+C$.
   • Fix: always write $\ln|x-2|+C$.

8. Improper integrals: evaluating like normal definite integrals

   • Wrong: plugging in the “bad” endpoint directly.
   • Fix: replace with a limit and check convergence.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can instantly recognize when to use basic rules vs $u$-sub vs IBP vs partial fractions.
• You know the must-have antiderivatives (power/log/exp/trig/inverse trig) and the signs.
• For $u$-sub, you rewrite **everything** in $u$ (and handle bounds correctly).
• For IBP, you use $\int u\,dv = uv - \int v\,du$ and choose $u$ using LIATE.
• For rational functions, you long divide first if needed, then partial fractions.
• For trig powers, you use odd/even strategies and half-angle when both even.
• For improper integrals, you convert to a limit and decide converge/diverge.
• For numerical methods, you can compute $T_n$ and $S_n$ and remember Simpson needs **even** $n$.

You’ve got the tools—now it’s just recognition + clean algebra.