Standard integrals are pre-computed antiderivatives that eliminate the need to derive each integral from first principles during exams.
## Core concept
A standard integral is the reverse of a standard derivative. Where differentiation gives a rate of change, integration recovers the original function (plus a constant). During exams, you apply memorised standard integral formulas directly rather than integration by parts or substitution for simple cases.
Key principle: If `d/dx[f(x)] = g(x)`, then `∫g(x)dx = f(x) + C`, where C is the constant of integration (required for indefinite integrals).
## Standard integral formulas
These must be memorised for the exam:
| Integrand | Standard Integral | Condition | |-----------|-------------------|-----------| | `x^n` | `x^(n+1)/(n+1) + C` | n ≠ −1 | | `1/x` or `x^(−1)` | `ln\|x\| + C` | x ≠ 0 | | `e^x` | `e^x + C` | Always | | `a^x` | `a^x/ln(a) + C` | a > 0, a ≠ 1 | | `sin(x)` | `−cos(x) + C` | x in radians | | `cos(x)` | `sin(x) + C` | x in radians | | `sec²(x)` | `tan(x) + C` | cos(x) ≠ 0 | | `cosec²(x)` | `−cot(x) + C` | sin(x) ≠ 0 | | `sec(x)tan(x)` | `sec(x) + C` | cos(x) ≠ 0 | | `cosec(x)cot(x)` | `−cosec(x) + C` | sin(x) ≠ 0 | | `1/(1+x²)` | `tan⁻¹(x) + C` or `arctan(x) + C` | Always | | `1/√(1−x²)` | `sin⁻¹(x) + C` or `arcsin(x) + C` | \|x\| < 1 |
## Rules for applying standard integrals
- Linearity: `∫[af(x) + bg(x)]dx = a∫f(x)dx + b∫g(x)dx` (a, b constants)
- Constant of integration: Always add `+ C` for indefinite integrals
- Power rule adjustment: When integrand is `x^n`, use the formula; for `(ax + b)^n`, use substitution or the chain rule variant
- Scaling: `∫f(kx)dx = (1/k)F(kx) + C` where F is the antiderivative of f and k is a constant
## Common exam applications
- Finding antiderivatives: Given a derivative, state the original function
- Area under curves: Apply definite integral notation using standard formulas
- Revenue/cost problems: Integrate marginal cost or revenue to get total cost/revenue
- Physics applications: Integrate acceleration to find velocity or displacement
## Worked example
Question: Find `∫(3x² + 2sin(x) − 1/x)dx`
Solution: - Separate by linearity: `3∫x²dx + 2∫sin(x)dx − ∫(1/x)dx` - Apply standard integrals: - `3 · x³/3 = x³` - `2 · (−cos(x)) = −2cos(x)` - `−ln|x|` - Answer: `x³ − 2cos(x) − ln|x| + C`
## Common mistakes
- Forgetting the constant C: Never write an indefinite integral answer without `+ C`
- Misapplying power rule: `∫(1/x)dx ≠ −1/x + C`; it equals `ln|x| + C`
- Ignoring domain restrictions: e.g. `∫1/√(1−x²)dx` valid only when |x| < 1
- Confusing trigonometric integrals: sin/cos signs matter—memorise exactly which integral gives −cos vs +sin
- Neglecting coefficients in scaled integrands: `∫cos(2x)dx ≠ sin(2x) + C`; it equals `(1/2)sin(2x) + C`
Practice these formulas repeatedly before the exam. Speed and accuracy with standard integrals directly impacts marks on integration questions.