Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) f(x) dx is the definite integral from x = a to x = b.
Recap
- Some integrands only match a rule after you rewrite them in an equivalent form. The rewriting is algebra; the function’s values do not change.
- Compare degrees first. If the top’s degree is at least the bottom’s, use long division: quotient + remainder/divisor. Check by multiplying back.
- For 1/(quadratic with no real roots), complete the square and use the arctan form.
- For 1/√(quadratic with a negative x² term), complete the square and use the arcsin form.
- For (linear)/(quadratic), split the top into k × (derivative of the bottom) + constant. The first part gives ln; the second gives arctan.
- If the denominator factors into linear factors (for example x² − 4), you need partial fractions: BC only, Topic 6.12.
Key relationships
| Situation | Rewrite | Result |
|---|---|---|
| Top-heavy rational function | Long division | Polynomial terms + a ln or arctan term |
| Completing the square | x² + bx + c = (x + b/2)² + (c − b²/4) | Sum of squares when b² − 4c < 0 |
| Arctan form (a > 0) | ∫ du/(a² + u²) | (1/a) arctan(u/a) + C |
| Arcsin form (a > 0) | ∫ du/√(a² − u²) | arcsin(u/a) + C |
| Derivative over function | ∫ g′(x)/g(x) dx | ln |
Assumptions behind the method
- The rewritten form must equal the original for every x in the interval (check by expanding or multiplying back).
- The interval of integration must avoid zeros of the denominator, and for arcsin it must keep the expression under the root positive.
- In the arctan and arcsin forms, a is the positive square root of the constant, not the constant itself.
Mistakes to avoid
- Integrating top and bottom separately.
- Dropping the remainder after long division.
- Using a² where a belongs, for example 1/4 instead of 1/2 in front of arctan when the constant is 4.
- Sign slips in (x − h)²: expand to check.
- Writing ln(quadratic) when the top is not a multiple of the bottom’s derivative.
- Using arctan for a quadratic that factors.
Quick self-check
- Find ∫ x²/(x + 1) dx. (x²/2 − x + ln|x + 1| + C, because x²/(x + 1) = x − 1 + 1/(x + 1))
- Find ∫ 1/(x² + 2x + 10) dx. ((1/3) arctan((x + 1)/3) + C, because x² + 2x + 10 = (x + 1)² + 9)
- Find ∫ 1/√(3 + 2x − x²) dx. (arcsin((x − 1)/2) + C, because 3 + 2x − x² = 4 − (x − 1)²)
Next: practice questions.