Calculus – Integrations (Copy)
CALCULUS – INTEGRATION — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Integration | ∫f(x)dx | Forgetting constant |
| Constant of Integration | + C | Omitting completely |
| Reverse Power Rule | ∫xⁿdx = xⁿ⁺¹/(n + 1) + C | Forgetting +1 |
| Special Rule | ∫1/x dx = lnx + C | Using power rule |
| Constant Integration | ∫k dx = kx + C | Writing k only |
| Sum Rule | ∫(u + v)dx = ∫u dx + ∫v dx | Integrating together incorrectly |
| Difference Rule | ∫(u − v)dx = ∫u dx − ∫v dx | Sign mistakes |
| Constant Multiple Rule | ∫kf(x)dx = k∫f(x)dx | Ignoring coefficient |
| Exponential Integration | ∫eˣdx = eˣ + C | Adding denominator |
| Trig Integration | ∫cosx dx = sinx + C | Wrong sign |
| Trig Integration | ∫sinx dx = −cosx + C | Forgetting negative |
| Definite Integral | ∫ₐᵇf(x)dx = F(b) − F(a) | Wrong substitution order |
| Area Under Curve | Use definite integral | Using coordinates only |
| Area Between Curves | ∫(top − bottom)dx | Reversing subtraction |
| Area Below x-axis | Take positive value | Leaving negative area |
| Velocity to Displacement | Integrate velocity | Differentiating instead |
| Acceleration to Velocity | Integrate acceleration | Differentiating instead |
| Differential Equation | Integrate derivative | Leaving derivative form |
| Initial Conditions | Substitute to find C | Leaving constant unknown |
| Polynomial Integration | Increase power by 1 then divide | Forgetting division |
| Fractional Powers | Apply power rule carefully | Wrong denominator |
| Negative Powers | Add 1 to index carefully | Dividing by zero issue |
| Special Case | ∫x⁻¹dx = lnx + C | Using normal power rule |
| Integration by Substitution | Replace variable carefully | Forgetting dx conversion |
| Definite Integral Without + C | Constants cancel | Adding unnecessary constants |
| Indefinite Integral | Must include + C | Forgetting entirely |
| Exact Area | Leave exact values | Decimalising too early |
| Graph Area Interpretation | Area = integral | Confusing with gradient |
| Kinematics Integration | Integrate step-by-step | Mixing variables |
| Area from Velocity-Time Graph | Integral gives displacement | Thinking distance automatically |
| Negative Velocity | Signed area | Ignoring negative sections |
| Reversing Integration | Integration is reverse differentiation | Using wrong power |
| Logarithmic Integration | ∫1/x dx = lnx + C | Writing log₁₀x |
| Exponential Growth Integration | ∫aˣdx = aˣ/lna + C | Forgetting denominator |
| Limits of Integration | Apply upper then lower | Reversing order |
| Simplifying Before Integrating | Expand if easier | Integrating complicated form directly |
| Units in Area | Integral gives units² sometimes | Ignoring units |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK INTEGRATION RULES TABLE
| Function | Integral |
|---|---|
| xⁿ | xⁿ⁺¹/(n + 1) + C |
| 1/x | lnx + C |
| eˣ | eˣ + C |
| aˣ | aˣ/lna + C |
| cosx | sinx + C |
| sinx | −cosx + C |
| k | kx + C |
QUICK DEFINITE INTEGRAL RULES
| Situation | Formula |
|---|---|
| Definite Integral | F(b) − F(a) |
| Area Under Curve | ∫ₐᵇf(x)dx |
| Area Between Curves | ∫(top − bottom)dx |
| Area Below x-axis | Use positive value |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK KINEMATICS RULES
| Quantity | Formula |
|---|---|
| Velocity | v = ds/dt |
| Acceleration | a = dv/dt |
| Displacement | ∫v dt |
| Velocity from Acceleration | ∫a dt |
QUICK AREA RULES
| Situation | Meaning |
|---|---|
| Positive area | Above x-axis |
| Negative area | Below x-axis |
| Total area | Add absolute values |
| Signed area | Include negatives |
SPECIAL CASES TABLE
| Expression | Integral |
|---|---|
| x⁻¹ | lnx + C |
| x¹⁄² | 2x³⁄²/3 + C |
| x⁻¹⁄² | 2x¹⁄² + C |
| sec²x | tanx + C |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Integrating powers | Add 1 then divide |
| Definite integral | Upper minus lower |
| Area below axis | Make positive at end |
| Differential equation | Integrate both sides |
| Initial conditions | Find C immediately |
| Complicated bracket | Expand first |
| x⁻¹ term | Use lnx rule |
| Trig integration | Memorise sign carefully |
| Kinematics | Integrate stepwise |
| Exact answers | Keep fractions and π exact |
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Forgetting + C | Incomplete integration |
| Wrong power after integration | Entire answer incorrect |
| Dividing incorrectly | Algebra errors |
| Using power rule on x⁻¹ | Invalid mathematics |
| Wrong trig sign | Formula marks lost |
| Reversing limits | Negative answers |
| Leaving negative area | Incorrect total area |
| Missing initial conditions | Unknown constant remains |
| Decimal rounding too early | Accuracy lost |
| Not simplifying final answer | Final marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.