Q99 / 160 · AP EAMCET 2020 (17 Sep Shift 2) · Mathematics

\(\int_{-\pi / 4}^{\pi / 4} x^3 \sin ^4(x) d x=\)

  • Correct answer
    A
    0
  • B
    \(\pi\)
  • C
    1
  • D
    \(2 \pi\)

Solution

\(I=\int_{-\pi / 4}^{\pi / 4} x^3 \cdot \sin ^4 x d x\) As, \(\int_{-a}^a f(x) d x=0\), when \(f(-x)=-f(x),\) We have, \(\begin{gathered} f(-x)=(-x)^3 \sin ^4(-x) \\ =-\left(x^3 \cdot \sin ^4 x\right)=-f(x) \\ \therefore \quad \int_{-\pi / 4}^{\pi / 4} x^3 \cdot \sin ^4 x \cdot d x=0 . \end{gathered}\)

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

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