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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\int\limits_{ - 3\pi }^{3\pi } {{{\sin }^2}\,\theta {\mkern 1mu} si{n^2}\,2\,\theta d\theta } $ is equal to 
  • A
    $\pi $
  • ✓
    $\frac{{3\pi }}{2}$
  • C
    $\frac{{5\pi }}{2}$
  • D
    $6\pi $
✓

Answer

Correct option: B.
$\frac{{3\pi }}{2}$
b
$\int_{-3 \pi}^{3 \pi} \sin ^{2} \theta \sin ^{2} 2 \theta \mathrm{d} \theta=2 \int_{0}^{3 \pi} \sin ^{2} \theta \sin ^{2} 2 \theta \mathrm{d} \theta$

$=8 \int_{0}^{3 \pi} \sin ^{4} \theta \cos ^{2} \theta \mathrm{d} \theta=24 \int_{0}^{\pi} \sin ^{4} \theta \cos ^{2} \theta \mathrm{d} \theta$

$=48 \int_{0}^{\pi / 2} \sin ^{4} \theta \cos ^{2} \theta d \theta$

$=\frac{48 \cdot(3.1) \cdot(1)}{6.4 \cdot 2} \cdot \frac{\pi}{2}=\frac{3 \pi}{2}$

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