Short on time? This page is the recap. For explanations, the family-of-curves figure and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- ∫ f(x) dx (no limits) is the indefinite integral: all antiderivatives of f.
- If F′ = f, then ∫ f(x) dx = F(x) + C. Antiderivatives on an interval differ only by a constant, so this lists them all.
- Graphically, the family is one curve shifted up and down. All members have the same slope at each x.
- Constant multiples and sums integrate term by term. Use one + C at the end.
- No product or quotient rule: expand, split or use an identity first.
- Some functions, such as e^(x²) and sin(x²), have no antiderivative you can write as a formula.
Key relationships
| Integral | Result |
|---|---|
| ∫ xⁿ dx, n ≠ −1 | xⁿ⁺¹/(n + 1) + C |
| ∫ (1/x) dx | ln|x| + C |
| ∫ eˣ dx; ∫ aˣ dx | eˣ + C; aˣ/ln a + C |
| ∫ cos x dx; ∫ sin x dx | sin x + C; −cos x + C |
| ∫ sec²x dx; ∫ csc²x dx | tan x + C; −cot x + C |
| ∫ sec x tan x dx; ∫ csc x cot x dx | sec x + C; −csc x + C |
| ∫ 1/√(1 − x²) dx; ∫ 1/(1 + x²) dx | arcsin x + C; arctan x + C |
Assumptions behind the rules
- Each result holds on an interval where the integrand is defined. For example, ln|x| + C works on x > 0 or on x < 0, not across 0.
- The constant C is arbitrary. A known point on the graph (Topic 7.7) is needed to fix it. In a definite integral it cancels, so it does not matter there.
Mistakes to avoid
- No + C on an indefinite integral.
- Power rule on 1/x, which divides by 0.
- ln for every fraction: ∫ 1/x² dx = −1/x + C, not ln(x²).
- Factor-by-factor integration of a product.
- Trig sign slips: ∫ sin x dx = −cos x + C; ∫ csc²x dx = −cot x + C.
- No dx, or an “=” between a function and its integral.
Quick self-check
- Find ∫ x^(2/3) dx. ((3/5)x^(5/3) + C)
- Find ∫ 7 dx. (7x + C)
- Find ∫ csc²x dx. (−cot x + C, because d/dx cot x = −csc²x)
- Is ∫ (x + 1)² dx = (x + 1)³/3 + C correct? (Yes: its derivative is (x + 1)². Expanding to x² + 2x + 1 and integrating gives x³/3 + x² + x + C, which differs only by the constant 1/3.)
Next: practice questions.