Q64 / 170 · Manipal (MET) 2017 · Mathematics

$\int_0^1 x(1-x)^{12} d x$ is equal to

  • A
    $\frac{1}{132}$
  • B
    $\frac{1}{156}$
  • Correct answer
    C
    $\frac{1}{182}$
  • D
    None of the above

Solution

$\begin{aligned} & \int_0^1 x(1-x)^{12} d x \\ = & {\left[\frac{x(1-x)^{13}}{-13}+\int_0^1 \frac{(1-x)^{13}}{13} d x\right]_0^1 } \\ = & {\left[\frac{x(1-x)^{13}}{-13}\right]_0^1+\left[\frac{(1-x)^{14}}{-14 \times 13}\right]_0^1 } \\ = & {[0-0]+\left[0-\frac{1}{-14 \times 13}\right]=\frac{1}{182} }\end{aligned}$

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

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