Q244 / 270 · NDA 2018 (Phase 1) · Mathematics

What is $\int_0^1 x(1-x)^9 d x$ equal to?

  • Correct answer
    A
    $\frac{1}{110}$
  • B
    $\frac{1}{132}$
  • C
    $\frac{1}{148}$
  • D
    $\frac{1}{240}$

Solution

$\begin{aligned} & \int_0^1 x(1-x)^9 \cdot d x \\ & =\int_0^1(1-x)(1-(1-x))^9 \cdot d x\left(\because \int_0^a f(x) \cdot d x=\int_0^a f(a-x) \cdot d x\right) \\ & =\int_0^1(1-x) \cdot x^9 \cdot d x=\int_0^1\left(x^9-x^{10}\right) \cdot d x \\ & =\left(\frac{x^{10}}{10}-\frac{x^{11}}{11}\right)_0^1=\frac{1}{10}-\frac{1}{11}=\frac{11-10}{110}=\frac{1}{110}\end{aligned}$

Hence, the correct answer is option: 'A'.

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