Question:
The value of
\[
\int_0^1 x(1-x)^9\,dx
\]
is
The value of
\[
\int_0^1 x(1-x)^9\,dx
\]
is
Show Hint
Use beta function for integrals of the form \(\int_0^1 x^a(1-x)^b\,dx\).
Updated On: May 7, 2026
- \(\frac{1}{110}\)
- \(\frac{1}{120}\)
- \(-\frac{1}{110}\)
- \(-\frac{1}{120}\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Use beta function: \[ \int_0^1 x^{m-1}(1-x)^{n-1}\,dx=B(m,n) \]
Step 2: Here: \[ x(1-x)^9=x^{2-1}(1-x)^{10-1} \] So, \[ m=2,\qquad n=10 \]
Step 3: \[ B(2,10)=\frac{\Gamma(2)\Gamma(10)}{\Gamma(12)} \]
Step 4: \[ =\frac{1!\,9!}{11!} \] \[ =\frac{1}{11\cdot 10} \] \[ =\frac{1}{110} \] \[ \boxed{\frac{1}{110}} \]
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