Assertion (A) : _0^ 2 ^3 x d x=0 Reason (R) : ^3 x is an odd function.

(b) : Let I= _0^ 2 ^3 x d x= _0^ 2 (1- ^2 x ) x d x Putting x=t x d x=-d t When x=0, t=1 and x=2 , t=1 I= _1^1 (1-t^2 )(-d t)=0

Assertion (A) : _0^ 2 ^3 x d x=0 Reason (R) : ^3 x is an odd func…

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Question

Assertion (A) : $\int_0^{2 \pi} \sin ^3 x d x=0$
Reason (R) : $\sin ^3 x$ is an odd function.

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Answer

(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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