Question

The question is in the screenshot

Answer 1

`\large \bf \implies[0, \pi]`

Explanation:

`\bf\longrightarrow{F(x) = \displaystyle \int\limits_(0)^(x) \sqrt{4 - {t}^(2) } \: \: dt}`

`\bf\longrightarrow{F(x) = \displaystyle \bigg[ (t)/(2) \sqrt{ {2}^(2) - {t}^(2) } + \frac{ {2}^(2) }{2} \sin^( - 1) (t)/(2) \bigg] _(0)^(x)}`

`\bf\longrightarrow{{F(x) = \displaystyle \bigg[ (x)/(2) \sqrt{ 4 - {x}^(2) } + 2\sin^( - 1) (x)/(2) - 0 - 0 \bigg]} }`

`\bf\longrightarrow{F(x) = \displaystyle (x)/(2) \sqrt{ 4 - {x}^(2) } + 2\sin^( - 1) (x)/(2)}`

• Since n ≤ 2

• So, maximum value of
  `F(x) = \displaystyle 2\sin^(-1)(2)/(2) + 0 = 2 (\pi)/(2) = \pi`

So, Correct option is (b) [0,π]