NCERT Class 12 · Ch 7 Exercise 7.10 · Q7
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.10 Question 7
Evaluate \(\int\limits_{0}^{1} x \left(1 - x\right)^{n}\, dx\) using definite-integral properties.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\frac{1}{\left(n + 1\right) \left(n + 2\right)}\)
- For n>−1 the endpoint integral converges. Put u=1−x, so I=∫₀¹(1−u)uⁿdu.
- Integrate the two powers: I=1/(n+1)−1/(n+2)=1/[(n+1)(n+2)]. If n is restricted to nonnegative integers this condition is automatic.
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