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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\mathop \smallint \limits_{\frac{{ - 3\pi }}{2}}^{\frac{{ - \pi }}{2}} \left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x + 3\pi } \right)} \right]dx = $
  • A
    $\frac{{{\pi ^4}}}{{32}}$
  • B
    $\;\frac{{{\pi ^4}}}{{32}}$+$\frac{\pi }{2}$
  • ✓
    $\;\frac{\pi }{2}$
  • D
    $\;\frac{\pi }{4} - 1$
✓

Answer

Correct option: C.
$\;\frac{\pi }{2}$
c
$I=\int_{-\frac{3 \pi}{2}}^{-\frac{\pi}{2}}\left[(x+\pi)^{3}+\cos ^{2}(x+3 \pi)\right] d x$

Put $x+\pi=t$

$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(t^{3}+\cos ^{2} t\right) d t$

$=2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos ^{2} t d t$

[using the property of even and odd function $]$

$=\int_{0}^{\frac{\pi}{2}}(1+\cos 2 t) d t=\frac{\pi}{2}+0$

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