Class 12 Integrals · Short Answer2 Marksdefinite_integration
Class 12 Maths · Integrals · Short Answer #50
For real n>−1, evaluate \(\int_0^1x(1-x)^n dx\).
✅ Step-by-Step Solution
Final Answer: 1/[(n+1)(n+2)]
- Set u=1−x; the integral becomes ∫₀¹(1−u)uⁿdu=∫₀¹(uⁿ−u^(n+1))du.
- Because n>−1 both terms converge, giving 1/(n+1)−1/(n+2)=1/[(n+1)(n+2)].
- Therefore the integral equals 1/[(n+1)(n+2)].
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Set u=1−x; the integral becomes ∫₀¹(1−u)uⁿdu=∫₀¹(uⁿ−u^(n+1))du.
- [1 mark] Because n>−1 both terms converge, giving 1/(n+1)−1/(n+2)=1/[(n+1)(n+2)].
- [0.5 marks] Therefore the integral equals 1/[(n+1)(n+2)].
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