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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
If $n$ is any integer, then $\int_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}(2n + 1)x\,dx = } $
  • A
    $x$
  • B
    $1$
  • ✓
    $0$
  • D
    None of these
✓

Answer

Correct option: C.
$0$
c
(c) Since $\cos (2n + 1)(\pi - x) = \cos [(2n + 1)\pi $$ - (2n + 1)x]$

$=  - \cos (2n + 1)x$ and ${\cos ^2}(\pi - x) = {\cos ^2}x$

So that $f(2a - x) = - f(x)$, and 

hence by the property of definite integral 

$\int_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}(2n + 1)x\,dx = 0} $.

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