Example 1 · level 1
Evaluate ∫ from 0 to 3 of (x + 2) dx.
Show solution
- F(x) = (1/2)x² + 2x.
- F(3) = 21/2 and F(0) = 0.
- Integral = F(3) − F(0) = 21/2.
Answer: 21/2
CLASS 12 MATHS · INTEGRALS
Definite integrals of polynomials
∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f. For xⁿ, the antiderivative is xⁿ⁺¹/(n + 1).
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Worked examples
Example 2 · level 2
Evaluate ∫ from 1 to 3 of (−3x² + 3x + 6) dx.
Show solution
- F(x) = −x³ + (3/2)x² + 6x.
- F(3) = 9/2 and F(1) = 13/2.
- Integral = F(3) − F(1) = −2.
Answer: -2
Example 3 · level 3
Evaluate ∫ from 0 to 3 of (3x³ + 4x² − 5x − 3) dx.
Show solution
- F(x) = (3/4)x⁴ + (4/3)x³ − (5/2)x² − 3x.
- F(3) = 261/4 and F(0) = 0.
- Integral = F(3) − F(0) = 261/4.
Answer: 261/4