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.