^ 2 _0x dx is equal to: 1. 4 2. 2 3. 4. 1 - Vidyadip

Question

$\int\limits^{\frac{\pi}{2}}_0\text{x}\sin\text{x dx}$ is equal to:

$\frac{\pi}{4}$
$\frac{\pi}{2}$
$\pi$
$1$

✓

Answer

1

Solution:
We have,
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\text{x}\sin\text{x dx}$
$=\big[-\text{x}\cos\text{x}\big]^{\frac{\pi}{2}}_0-\int\limits^{\frac{\pi}{2}}_01(-\cos\text{x})\text{dx}$
$=\big[-\text{x}\cos\text{x}\big]^{\frac{\pi}{2}}_0+\int\limits^{\frac{\pi}{2}}_0\cos\text{x dx}$
$=-\big[\text{x}\cos\text{x}\big]^{\frac{\pi}{2}}_0+\big[\sin\text{x}\big]^{\frac{\pi}{2}}_0$
$=-\big[0-0\big]+\big[1-0\big]$
$=1$

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