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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Assertion (A) : $\int_0^{2 \pi} \sin ^3 x d x=0$
Reason (R) : $\sin ^3 x$ is an odd function.
  • A
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • ✓
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • D
    (A) is false but (R) is true.
✓

Answer

Correct option: B.
Both (A) and (R) are true but (R) is not the correct explanation of (A).
(b) : Let $I=\int_0^{2 \pi} \sin ^3 x d x=\int_0^{2 \pi}\left(1-\cos ^2 x\right) \sin x d x$
Putting $\cos x=t \Rightarrow \sin x d x=-d t$
When $x=0, t=1$ and $x=2 \pi, t=1$
$\therefore \quad I=\int_1^1\left(1-t^2\right)(-d t)=0$

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