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(D) $(i)$ Let $I_1 = \int_0^{\pi / 2} \sin ^m x \cos x d x$.Substitute $\sin x = t$,then $\cos x d x = d t$.When $x = 0, t = 0$ and when $x = \pi / 2,

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Both $(i)$ and (ii) are true

Vedclass · Answer

(D) $(i)$ Let $I_1 = \int_0^{\pi / 2} \sin ^m x \cos x d x$.Substitute $\sin x = t$,then $\cos x d x = d t$.When $x = 0, t = 0$ and when $x = \pi / 2, t = 1$.$I_1 = \int_0^1 t^m d t = \left[ \frac{t^{m+1}}{m+1} \right]_0^1 = \frac{1}{m+1} (1^{m+1} - 0^{m+1}) = \frac{1}{m+1}$.Thus,statement $(i)$ is true.(ii) Let $I_2 = \int_0^{\pi / 2} \sin x \cos ^n x d x$.Substitute $\cos x = t$,then $-\sin x d x = d t$,or $\sin x d x = -d t$.When $x = 0, t = 1$ and when $x = \pi / 2, t = 0$.$I_2 = \int_1^0 t^n (-d t) = \int_0^1 t^n d t = \left[ \frac{t^{n+1}}{n+1} \right]_0^1 = \frac{1}{n+1} (1^{n+1} - 0^{n+1}) = \frac{1}{n+1}$.Thus,statement (ii) is true.Therefore,both $(i)$ and (ii) are true.

Choose the correct option regarding the following definite integrals:
$(i)$ $\int_0^{\pi / 2} \sin ^m(x) \cos (x) d x = \frac{1}{m+1}$
(ii) $\int_0^{\pi / 2} \sin (x) \cos ^n(x) d x = \frac{1}{n+1}$

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Choose the correct option regarding the following definite integrals:$(i)$ $\int_0^{\pi / 2} \sin ^m(x) \cos (x) d x = \frac{1}{m+1}$(ii) $\int_0^{\pi / 2} \sin (x) \cos ^n(x) d x = \frac{1}{n+1}$