Choose the correct option regarding the follo | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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,

Vedclass · Accepted Answer

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) $\int_0^{\pi / 2} \sin (x) \cos ^n(x) d x = \frac{1}{n+1}$

Similar Questions

$\int_0^{\pi /4} {\frac{{\sin x + \cos x}}{{9 + 16\sin 2x}}\,dx = } $

$\int_0^\pi |\cos x| \, dx = $

Evaluate the definite integral $\int_{-1}^{2} \left[ \frac{[x]}{1 + x^2} \right] dx$,where $[\cdot]$ denotes the Greatest Integer Function $(GIF)$.

$\int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx = $

On the interval $\left[ \frac{5\pi}{3}, \frac{7\pi}{4} \right]$,the greatest value of the function $f(x) = \int_{5\pi/3}^x (6\cos t - 2\sin t) \, dt$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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) $\int_0^{\pi / 2} \sin (x) \cos ^n(x) d x = \frac{1}{n+1}$