How do you integrate a polynomial?
Short answer
Integrate each term separately: for axⁿ, add one to the power and divide by the new power, giving axⁿ⁺¹/(n + 1). Always add a constant of integration, + c, for an indefinite integral. For example, ∫(6x² + 4x) dx = 2x³ + 2x² + c. Integration is the reverse of differentiation.
Step by step
- Write every term as a power of x, rewriting roots and fractions first.
- Add 1 to each power.
- Divide each term by its new power.
- Add + c for an indefinite integral.
- For a definite integral, substitute the upper limit, then the lower limit, and subtract. No + c is needed.
Worked example
Find the area under y = 3x² from x = 1 to x = 2
- ∫3x² dx = x³
- Upper limit: 2³ = 8
- Lower limit: 1³ = 1
- Area = 8 − 1 = 7 square units
Common mistakes
- Forgetting + c.
- Multiplying by the power instead of dividing, as if differentiating.
- Subtracting the limits in the wrong order.
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