Q90 / 170 · Manipal (MET) 2018 · Mathematics

$\int_0^1 x \sin \pi x d x$ is equal to

  • A
    1
  • B
    1/2
  • C
    $\pi$
  • Correct answer
    D
    $1 / \pi$

Solution

$\begin{array}{rl}\int_0^1 & x \sin \pi x d x \\ \quad & =\left[-x \frac{\cos \pi x}{\pi}\right]_0^1+\int_0^1 \frac{\cos \pi x}{\pi} d x \\ \quad=\left[-x \frac{\cos \pi x}{\pi}+\frac{\sin \pi x}{\pi^2}\right]_0^1=\left[\frac{-(-1)}{\pi}\right]=\frac{1}{\pi}\end{array}$

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

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