Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #86
Evaluate \(\int_{-1}^2|x^3-x|dx\).
![Absolute-value integral on [−1,2]. Blue: x³−x. Rose: |x³−x|. Split at −1, 0 and 1.](/data/assets/integrals-1/cubic-absolute-area.svg)
✅ Step-by-Step Solution
Final Answer: 11/4
- Factor x³−x=x(x−1)(x+1). Its signs on (−1,0),(0,1),(1,2) are +,−,+.
- For F=x⁴/4−x²/2, write the integral as [F]₋₁⁰−[F]₀¹+[F]₁².
- The three positive contributions are 1/4,1/4,9/4, summing to 11/4.
- Therefore the integral equals 11/4.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Factor x³−x=x(x−1)(x+1). Its signs on (−1,0),(0,1),(1,2) are +,−,+.
- [1 mark] For F=x⁴/4−x²/2, write the integral as [F]₋₁⁰−[F]₀¹+[F]₁².
- [1 mark] The three positive contributions are 1/4,1/4,9/4, summing to 11/4.
- [0.5 marks] Therefore the integral equals 11/4.
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