2pi-4pi-1-cos-x-dx-

Question Number 61801 by aliesam last updated on 08/Jun/19

\underset{2 π}{\int^{4 π}} \sqrt{1 - c o s (x)} d x

Commented by MJS last updated on 08/Jun/19

\sqrt{1 - \cos x} = \sqrt{2} \sin \frac{x}{2}

now {it}^{'} s easy

Commented by maxmathsup by imad last updated on 09/Jun/19

l e t A = \int_{2 π}^{4 π} \sqrt{1 - c o s x} d x c h a 7 g e m e n t x = 2 π + t g i v e

A = \int_{0}^{2 π} \sqrt{1 - c o s t} d t = \int_{0}^{2 π} \sqrt{2 {s i n}^{2} (\frac{t}{2})} d t = \sqrt{2} \int_{0}^{2 π} ∣ s i n (\frac{t}{2}) ∣ d t

0 ⩽ \frac{t}{2} ⩽ π \Rightarrow s i n (\frac{t}{2}) ⩾ 0 \Rightarrow A = \sqrt{2} \int_{0}^{2 π} s i n (\frac{t}{2}) = \sqrt{2} {[- 2 c o s (\frac{t}{2})]}_{0}^{2 π}

= - 2 \sqrt{2} {c o s (π) - c o s (0)} = - 2 \sqrt{2} (- 2) = 4 \sqrt{2}

A = 4 \sqrt{2} .

Commented by aliesam last updated on 09/Jun/19

t h a n k s s i r

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