AP Calculus BC Score Calculator - Raw Score to AP Grade Converter
Comprehensive AP Calculus BC score calculator with AB subscore calculation. Convert raw multiple-choice and free-response scores to final AP grades (1-5) using official College Board scoring guidelines. Includes composite score calculation and detailed performance analysis.
AP Calculus BC Score Calculator
Section I: Multiple Choice
Section II: Free Response
Understanding AP Calculus BC Scoring
AP Calculus BC is scored similarly to AB but with different content and curve adjustments. The exam includes all AP Calculus AB topics plus additional BC-only content (parametric equations, polar coordinates, vectors, and series). Students receive both a BC score (1-5) and an AB subscore (1-5) indicating their performance on AB topics within the BC exam. The scoring structure maintains the 50-50 split between multiple-choice and free-response sections.
AP Calculus BC Score Calculation Formula
Raw Score Calculation
Multiple Choice Raw Score:
\[ \text{MC Raw Score} = \text{Number Correct} \times 1.2 \]
45 questions × 1.2 multiplier = 54 points maximum
Free Response Raw Score:
\[ \text{FRQ Raw Score} = \sum_{i=1}^{6} \text{Points}_i \]
6 questions × 9 points each = 54 points maximum
Composite Score:
\[ \text{Composite Score} = \text{MC Raw} + \text{FRQ Raw} \]
Maximum: 54 + 54 = 108 points
AB Subscore Calculation
The AB subscore reflects performance on AB-level content within the BC exam:
\[ \text{AB Subscore Composite} = \text{AB-related MC} + \text{AB-related FRQ} \]
College Board identifies which questions test AB vs. BC-only topics and calculates a separate composite score, then converts to the AB subscore (1-5).
AP Calculus BC Score Conversion Table
| Composite Score Range | AP BC Score | Description | College Credit | % of Students |
|---|---|---|---|---|
| 65-108 | 5 | Extremely Well Qualified | Calc I & II credit | ~40% |
| 54-64 | 4 | Well Qualified | Calc I & often II | ~18% |
| 42-53 | 3 | Qualified | Calc I credit | ~20% |
| 30-41 | 2 | Possibly Qualified | Rarely grants credit | ~14% |
| 0-29 | 1 | No Recommendation | No credit | ~8% |
BC vs. AB: Key Differences
| Aspect | AP Calculus BC | AP Calculus AB |
|---|---|---|
| Content Coverage | All AB topics + series, parametric, polar, vectors | Limits, derivatives, integrals, differential equations |
| College Equivalent | Calculus I & II (2 semesters) | Calculus I (1 semester) |
| Score Distribution | ~40% earn 5s | ~20% earn 5s |
| Difficulty | More content, faster pace | Fewer topics, deeper focus |
| Typical Students | Strong math backgrounds, accelerated | Standard calculus preparation |
| Score for 5 | ~60-65% correct | ~63-65% correct |
| AB Subscore | Yes - included automatically | N/A |
What Score Do You Need?
| Target BC Score | Minimum Composite | MC Questions Needed | FRQ Points Needed | Percentage |
|---|---|---|---|---|
| 5 | ~65/108 | ~27/45 (60%) | ~32/54 (59%) | 60% |
| 4 | ~54/108 | ~23/45 (51%) | ~27/54 (50%) | 50% |
| 3 | ~42/108 | ~18/45 (40%) | ~21/54 (39%) | 39% |
| 2 | ~30/108 | ~13/45 (29%) | ~15/54 (28%) | 28% |
AP Calculus BC Topic Coverage
| Unit | Topic | Exam Weight | BC-Only Content |
|---|---|---|---|
| Units 1-8 | All AB Content | ~60% | No (AB subscore based on these) |
| Unit 9 | Parametric Equations, Polar Coordinates, Vector-Valued Functions | 11-12% | Yes - BC only |
| Unit 10 | Infinite Sequences and Series | 17-18% | Yes - BC only |
| Key BC Topics: Taylor/Maclaurin series, convergence tests, parametric derivatives, polar area, arc length | |||
Worked Examples
Example 1: High BC Score with Strong Performance
Multiple Choice:
- Part A (No Calculator): 27/30 correct
- Part B (Calculator): 14/15 correct
- Total MC: 41/45 correct
MC Raw Score: 41 × 1.2 = 49.2 points
Free Response:
- Q1: 8/9, Q2: 7/9, Q3: 8/9
- Q4: 7/9, Q5: 8/9, Q6: 7/9
- Total FRQ: 45/54 points
Composite Score: 49.2 + 45 = 94.2 ≈ 94
Final BC Score: 5 (Extremely Well Qualified)
Likely AB Subscore: 5
Example 2: Solid BC Performance
Multiple Choice: 28/45 correct
MC Raw Score: 28 × 1.2 = 33.6 points
Free Response: 32/54 points
Composite Score: 33.6 + 32 = 65.6 ≈ 66
Final BC Score: 5 (Extremely Well Qualified)
Note: BC cutoff for 5 is lower than AB (~60% vs ~63%)
Understanding the AB Subscore
What is the AB Subscore?
Every AP Calculus BC student automatically receives an AB subscore (1-5) in addition to their BC score. This subscore reflects performance on AB-level topics within the BC exam:
- Purpose: Shows mastery of Calculus I content for students who take BC
- Calculation: Based on subset of BC questions covering AB topics
- College use: Some colleges grant Calc I credit for AB subscore even if BC score doesn't meet requirements
- Typical relationship: AB subscore is often equal to or 1 point lower than BC score
Example scenarios:
- BC Score: 5, AB Subscore: 5 (mastered all content)
- BC Score: 4, AB Subscore: 4 or 5 (strong on AB, struggled with BC-only topics)
- BC Score: 3, AB Subscore: 4 (struggled with series/parametric but solid on AB content)
College Credit Comparison
| BC Score | AB Subscore | Typical Credit Granted | Course Placement |
|---|---|---|---|
| 5 | 5 | Calculus I & II (8 credits) | Skip to Calculus III / Multivariable |
| 4 | 4-5 | Calculus I & II (6-8 credits) | Skip to Calculus III (most schools) |
| 3 | 3-4 | Calculus I only (3-4 credits) | Start with Calculus II |
| 3 | 5 | Some schools: Calc I & II using subscore | Policy varies - check specific colleges |
| 2 | 3-4 | Sometimes Calc I via AB subscore | Rare - verify with admissions |
Common Misconceptions
BC is Not Just "Harder AB"
AP Calculus BC covers significantly more content than AB—approximately equivalent to two college semesters versus one. BC includes entire AB curriculum PLUS parametric equations, polar coordinates, vectors, and extensive series content (convergence tests, Taylor series, etc.). The BC exam isn't simply harder AB questions; it tests additional advanced topics. Students should choose BC if they have strong math backgrounds and can handle faster pacing, not just because they're "good at math."
Higher BC Score Percentages Don't Mean It's Easier
Approximately 40% of BC students earn 5s versus 20% for AB. This doesn't mean BC is easier—it reflects self-selection. Students choosing BC typically have stronger math backgrounds, more preparation, and greater motivation. The BC exam itself is more challenging with additional content. The generous scoring curve (60% for a 5 vs. AB's 63-65%) partially compensates for difficulty, but mainly reflects College Board's expectation that BC students demonstrate readiness for more advanced mathematics.
AB Subscore Isn't Always Equal to BC Score
While many students' AB subscores match their BC scores, they can differ by ±1-2 points. A student might score 4 on BC overall but 5 on AB subscore (mastered AB topics but struggled with series). Conversely, though rarer, BC score might exceed AB subscore if a student excels at BC-only topics. The subscore provides valuable information about strengths/weaknesses and can affect college credit decisions at schools with separate policies for BC score versus AB subscore.
Frequently Asked Questions
What percentage do you need for a 5 on AP Calc BC?
You typically need approximately 60-62% of total possible points to earn a 5 on AP Calculus BC. This translates to a composite score around 65-67 out of 108 points. In practical terms, you could answer 27-28 multiple-choice questions correctly (out of 45) and earn 32-35 points on free-response (out of 54) to achieve a 5. The BC cutoff is slightly lower than AB (which requires ~63-65%) because BC content is more extensive and challenging. The exact cutoff varies by year as College Board adjusts for exam difficulty.
Should I take AP Calc AB or BC?
Choose BC if you have strong math foundations, completed precalculus successfully (A/A- grades), enjoy mathematics, and can handle faster-paced learning. BC covers double the content in the same timeframe, equivalent to two college semesters. Choose AB if you want thorough coverage of Calculus I, prefer deeper exploration of fewer topics, or have average (not exceptional) precalculus performance. Both exams are rigorous and respected by colleges. Your decision should consider math strength, time availability for study, and college major plans (STEM majors benefit more from BC's advanced placement). Consult your math teacher—they know your capabilities and learning style.
What is the AB subscore on AP Calc BC?
The AB subscore is a secondary score (1-5) automatically calculated for all BC test-takers, reflecting performance on AB-level topics within the BC exam. College Board identifies which BC questions test AB content (limits, derivatives, integrals, applications, differential equations) versus BC-only material (series, parametric, polar). They calculate a separate composite score from AB-related questions and convert it to the AB subscore. This provides colleges additional information—some grant Calculus I credit based on AB subscore even if BC score doesn't meet their threshold. Most students' AB subscores equal or slightly differ from their BC scores.
How hard is it to get a 5 on AP Calc BC?
Earning a 5 on AP Calc BC requires solid understanding but is achievable with proper preparation. Approximately 40% of BC students earn 5s (compared to 20% for AB), though this reflects self-selection of stronger students choosing BC. You need ~60% correct overall—missing 18 multiple-choice questions and losing 20+ FRQ points still yields a 5. Success requires: mastering all AB content thoroughly, understanding series convergence deeply, comfort with parametric/polar equations, strong algebra skills, effective time management, and extensive practice with past exams. The generous curve reflects BC's advanced content. With consistent study and good teaching, motivated students regularly achieve 5s.
Can you take AP Calc BC without taking AB?
Yes, you can take AP Calculus BC without taking AB—BC covers all AB content plus additional topics. Most students who take BC do so directly without taking AB first, either through accelerated courses or self-study. BC is designed as a comprehensive two-semester calculus course, not a sequel to AB. However, you need strong precalculus foundations and ability to learn quickly. Some schools offer AB then BC sequentially (less common), but most offer parallel tracks: choose either AB or BC based on preparation level. Taking AB first then BC creates unnecessary repetition and delays advanced mathematics. If qualified, go directly to BC.
Do colleges prefer AP Calc BC over AB?
Selective colleges and STEM programs generally view BC more favorably because it demonstrates greater mathematical achievement and covers more material. BC earns credit for two semesters of calculus at most schools versus AB's one semester, allowing faster progression to advanced mathematics. For engineering, physics, mathematics, and computer science majors, BC is often strongly recommended or expected. However, AB is still highly respected and sufficient for many majors. Non-STEM majors and less selective colleges view both equally. What matters most is performing well on whichever exam you take—a 5 on AB is better than a 3 on BC for admissions purposes. Choose based on your capabilities and preparation, not just perceived prestige.
Using This Calculator for Practice Tests
Maximize this calculator's value for exam preparation:
- Take full practice tests: Use released College Board exams under timed conditions
- Calculate scores immediately: See your performance while the test is fresh
- Track progress: Create a spreadsheet documenting scores from multiple practice exams
- Identify patterns: Note whether MC or FRQ needs more work; analyze section-specific weaknesses
- Set realistic goals: Determine target scores for MC and FRQ based on your goal AP score
- Adjust study plans: Focus on topics where you consistently lose points
- Build confidence: Watch your scores improve with dedicated practice
Note: BC scoring curves vary by year. This calculator uses typical conversion ranges. Your actual score may differ by ±1-2 points depending on exam difficulty and College Board's equating process.
Important Disclaimer
This calculator provides estimated AP Calculus BC scores based on typical College Board conversion scales from recent exams. Actual score conversions vary by exam year as College Board adjusts for exam difficulty through statistical equating. Composite score cutoffs for each AP grade (1-5) can shift ±2-3 points between administrations. AB subscore calculations are estimated based on typical AB content within BC exams—actual subscores are calculated by College Board using proprietary methods. This tool is for educational planning and practice test analysis—official AP scores and AB subscores are determined solely by College Board. Use for study guidance and goal-setting, understanding that actual exam results may differ. For definitive scoring information and current conversion guidelines, consult College Board's AP Central website. This calculator does not replace official College Board scoring or guarantee any specific exam outcome.
AP Calculus BC – 2025 Cheatsheet
- Order of growth rates from fastest to slowest: \(x^x, x!, a^x, x^n, x\ln x, \ln x\)
- Methods to algebraically simplify limits if you can't directly plug in: Completing the square, Rationalization, Factoring
- Limit Properties:
- \(\lim_{x\to c}(af(x)) = a\lim_{x\to c}f(x)\)
- \(\lim_{x\to c}(f(x) \pm g(x)) = \lim_{x\to c}f(x) \pm \lim_{x\to c}g(x)\)
- \(\lim_{x\to c}(f(x)g(x)) = \lim_{x\to c}f(x) \cdot \lim_{x\to c}g(x)\)
- \(\lim_{x\to c}\frac{f(x)}{g(x)} = \frac{\lim_{x\to c}f(x)}{\lim_{x\to c}g(x)}\), provided \(\lim_{x\to c}g(x) \neq 0\)
- \(\lim_{x\to c}f(g(x)) = f(\lim_{x\to c}g(x))\), if f is continuous at \(\lim_{x\to c}g(x)\)
- Continuity exists if \(f(c) = \lim_{x\to c}f(x)\)
- Intermediate Value Theorem: Write "Since f(x) is continuous on [a,b] and f(c) is between f(a) and f(b), by the IVT there is a c in (a,b) such that f(c)=0"
- All differentiable functions are continuous, but not all continuous functions are differentiable
- Average Rate of Change = \(\frac{f(x+h)-f(x)}{h}\) or \(\frac{f(b)-f(a)}{b-a}\)
- Derivative Definition: \(f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{z\to x}\frac{f(z)-f(x)}{z-x}\)
- Power Rule: \(\frac{d}{dx}x^n = nx^{n-1}\)
- Sum/Difference Rule: \(\frac{d}{dx}[f(x)\pm g(x)] = f'(x)\pm g'(x)\)
- Product Rule: \(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
- Quotient Rule: \(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\)
- Common Derivatives:
- \(\frac{d}{dx}c = 0\) (c is constant)
- \(\frac{d}{dx}e^x = e^x\)
- \(\frac{d}{dx}kf(x) = kf'(x)\)
- \(\frac{d}{dx}\ln(x) = \frac{1}{x}\)
- \(\frac{d}{dx}\sin(x) = \cos(x)\)
- \(\frac{d}{dx}\cos(x) = -\sin(x)\)
- \(\frac{d}{dx}\tan(x) = \sec^2(x)\)
- \(\frac{d}{dx}\cot(x) = -\csc^2(x)\)
- \(\frac{d}{dx}\sec(x) = \sec(x)\tan(x)\)
- \(\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)\)
- Chain Rule: \(\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)\)
- \(\frac{d}{dx}a^x = a^x\ln(a)\)
- \(\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}\)
- Implicit Differentiation: Differentiate each term with respect to the individual variables, multiply by \(\frac{dy}{dx}\) when differentiating terms with y
- Derivatives of Inverse Trig Functions:
Function Derivative \(\sin^{-1}x\) \(\frac{1}{\sqrt{1-x^2}}\) \(\cos^{-1}x\) \(\frac{-1}{\sqrt{1-x^2}}\) \(\tan^{-1}x\) \(\frac{1}{1+x^2}\) \(\cot^{-1}x\) \(\frac{-1}{1+x^2}\) \(\sec^{-1}x\) \(\frac{1}{|x|\sqrt{x^2-1}}\) \(\csc^{-1}x\) \(\frac{-1}{|x|\sqrt{x^2-1}}\)
- Particle Motion:
- Position = \(s(t)\)
- Velocity = \(v(t) = s'(t)\)
- Acceleration = \(a(t) = v'(t) = s''(t)\)
- If velocity is negative, the particle is moving to the left.
- If velocity is positive, the particle is moving to the right.
- If velocity and acceleration have the same sign, the particle is speeding up.
- If velocity and acceleration have different signs, the particle is slowing down.
- Steps for Related Rates:
- Draw a picture and label the picture (assign variables)
- List your knowns and unknown values
- Write an equation to model the situation
- Take the derivative of both sides. Remember: \(\frac{d}{dt}\)
- Plug in known values and solve for desired values. DON'T FORGET UNITS!
- Linearization: \(f(c+a) \approx f(c) + f'(c)a\)
- L'Hopital's Rule: If \(\lim_{x\to c}\frac{f(x)}{g(x)}\) is indeterminate (\(0/0\) or \(\infty/\infty\)), then \(\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}\)
- Mean Value Theorem: Write "Since f(x) is continuous on [a,b] and differentiable on (a,b), there exists a c in (a,b) such that \(f'(c) = \frac{f(b)-f(a)}{b-a}\) by the MVT."
- Extreme Value Theorem: Write "Since f(x) is continuous on [a,b], by the EVT, there exists at least one local maximum and one local minimum on [a,b]."
- Critical Points: where \(f'(x) = 0\) or does not exist
- \(f'(x) > 0\): increasing; \(f'(x) < 0\): decreasing
- First Derivative Test: where \(f'(x) = 0\), if \(f'(x)\) changes from + to −: local max; if \(f'(x)\) changes from − to +: local min
- Determining Concavity: \(f''(x) > 0\): concave up; \(f''(x) < 0\): concave down; \(f''(x) = 0\): potential inflection point
- Second Derivative Test: If \(f'(x) = 0\), then:
- if \(f''(x) > 0\): local min
- if \(f''(x) < 0\): local max
- if \(f''(x) = 0\): test is inconclusive
- Steps for Optimization:
- Draw picture
- Label your picture and assign variables
- Write an equation and use given information to find relationships among variables
- Find extrema (min/max) and evaluate the function
- The integral is the area between the graph and the x-axis
- Riemann Sum can be used to approximate area (includes left, right, midpoint, and trapezoidal sum)
- \(\int_a^b f(x)dx \approx \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta x\) where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + i\Delta x\)
- Fundamental Theorem of Calculus (FTC): If \(F'(x) = f(x)\), then \(\int_a^b f(x)dx = F(b) - F(a)\)
- Basic Integration Formulas (NEVER FORGET + C):
- \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) (n ≠ -1)
- \(\int \frac{1}{x} dx = \ln|x| + C\)
- \(\int e^x dx = e^x + C\)
- \(\int \sin(x) dx = -\cos(x) + C\)
- \(\int \cos(x) dx = \sin(x) + C\)
- \(\int \sec^2(x) dx = \tan(x) + C\)
- \(\int \csc^2(x) dx = -\cot(x) + C\)
- \(\int \sec(x)\tan(x) dx = \sec(x) + C\)
- \(\int \csc(x)\cot(x) dx = -\csc(x) + C\)
- Integration by Parts (IBP): \(\int u\,dv = uv - \int v\,du\)
- Learn the Tabular Method to make IBP easier
- Use Partial Fraction Decomposition to integrate rational functions
- Improper Integrals: \(\int_a^{\infty} f(x)dx = \lim_{b\to\infty} \int_a^b f(x)dx\)
- A slope field is a graphical representation of a differential equation in the form \(dy/dx = f(x,y)\)
- Logistic Differential Equation: \(\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)\) where P is the population, L is the carrying capacity, and k is a constant
- Euler's method can be used to find the numerical values of functions based on a given differential equation and an initial condition
- Average Value: \(\frac{1}{b-a} \int_a^b f(x)dx\)
- Average Value Theorem: Write "Since f(x) is continuous on [a,b], by the AVT, there must be a c in (a,b) where \(f(c) = \frac{1}{b-a} \int_a^b f(x)dx\)."
- Position and velocity relationship:
- Acceleration = \(a(t)\)
- Velocity = \(v(t) = \int a(t) dt\)
- Position = \(s(t) = \int v(t) dt\)
- Speed = \(|v(t)|\)
- Distance traveled = \(\int |v(t)| dt\)
- Volume using the Washer Method: \(\pi \int (R_{\text{outer}}^2 - R_{\text{inner}}^2) dx\)
- Volume using the Disc Method: \(\pi \int R^2 dx\)
- Arc Length: \(\int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}dx\)
- Second Derivative of Parametric Equation: \(\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}\)
- Arc Length for Parametric Functions: \(s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt\)
- Slope of a Tangent Line for Polar Equations: \(\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}\)
- Polar Conversions: \(x = r \cos\theta\), \(y = r \sin\theta\), \(r = \sqrt{x^2 + y^2}\)
- Area under Polar Curves: \(A = \int_\alpha^\beta \frac{1}{2}r^2 d\theta\)
- Area under two Polar Curves: \(A = \int_\alpha^\beta \frac{1}{2}(r_1^2 - r_2^2) d\theta\)
- Arc Length for Polar Functions: \(L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta\)
- Sequences: arithmetic and geometric
- Series:
- Harmonic series \(\sum \frac{1}{n}\) diverges
- Power Series with terms \(\frac{1}{n^p}\) converges when \(p > 1\), else it diverges
- Alternating Series: For an alternating series (terms change sign), converges if \(\lim_{n\to\infty} a_n = 0\), else it diverges
- Taylor Series: \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n\)
- Maclaurin Series (make sure you memorize them!): Taylor series centered at \(c = 0\)
- Various tests used to determine convergence and divergence:
- nth Term Test: If \(\lim_{n\to\infty} a_n \neq 0\), then the series diverges
- Limit Comparison Test
- Direct Comparison Test
- Integral Test
- Alternating Series Test
- Ratio Test
- Be sure to show ALL of your work (even when using a calculator).
- Work on all of the parts that you know first before moving onto other parts. (Get those easy points!)
- Do not round any values as you complete the problem! Wait all the way until the end to round your answer to 3 or 4 decimal places.
- You can use abbreviations like IVT (Intermediate Value Theorem), MVT (Mean Value Theorem), and FTOC (Fundamental Theorem of Calculus)
- Do not simplify your answers unless specified. You don't want to lose points on steps you don't need to do!
- Memorize your important theorems and convergence tests! You'll need to know the conditions where the theorems and tests are met.
- Keep an eye on the time and pace yourself.
- For problems involving derivatives, clearly identify the rule you're using (chain rule, product rule, etc.)
- For series problems, clearly state which convergence test you're applying and why
- When working with function values from a table, clearly indicate which values you're using and how you're using them
- Always include units in your final answer when appropriate (especially for related rates and optimization problems)
- Practice with past FRQs to get comfortable with the format and expectations
- Squeeze Theorem: If \(f(x) \le h(x) \le g(x)\) and \(\lim_{x\to c} f(x) = L = \lim_{x\to c} g(x)\), then \(\lim_{x\to c} h(x) = L\)
- Pythagorean Identities:
- \(\sin^2 x + \cos^2 x = 1\)
- \(1 + \tan^2 x = \sec^2 x\)
- \(1 + \cot^2 x = \csc^2 x\)
- Double Angle Formulas:
- \(\sin 2x = 2\sin x \cos x\)
- \(\cos 2x = \cos^2 x - \sin^2 x\)
- \(\cos 2x = 2\cos^2 x - 1\)
- \(\cos 2x = 1 - 2\sin^2 x\)
- Common Maclaurin Series:
- \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots\)
- \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\)
- \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\)
- \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (-1 < x \le 1)\)
- \(\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots \quad (|x| < 1)\)
- Integration Substitution Techniques:
- For integrals of the form \(\int f(ax+b) dx\), use \(u = ax+b\)
- For integrals containing \(\sqrt{a^2-x^2}\), try \(u = a \sin \theta\) or \(u = a \cos \theta\)
- For integrals containing \(\sqrt{a^2+x^2}\), try \(u = a \tan \theta\)
- For integrals containing \(\sqrt{x^2-a^2}\), try \(u = a \sec \theta\)