-3-2 -1-cos x-1-cos x5-2dx- -AI CBSE 1980-

The correct option is B I=∫_π/3^π/2√(1+cos x)/(1 cos x)^5/2×√(1 cos x)/1 cos xdx =∫_π/3^π/2sin x/(1 cos x)^3dx Now, put 1 cos x=t Also, when x=π/3,t=1/2 and ...

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Standard XII

Mathematics

Property 1

∫π/3π/2 √1+co...

Question

$\int_{\frac{π}{3}}^{\frac{π}{2}} \frac{\sqrt{1 + c o s x}}{(1 - c o s x)^{\frac{5}{2}}} d x =$ [AI CBSE 1980]

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Solution

The correct option is B

$I = \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{\sqrt{1 + c o s x}}{(1 - c o s x)^{\frac{5}{2}}} \times \frac{\sqrt{1 - c o s x}}{1 - c o s x} d x$
$= \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{s i n x}{(1 - c o s x)^{3}} d x$
Now, put 1 - cos~x=t
Also, when $x = \frac{π}{3}, t = \frac{1}{2} a n d x = \frac{π}{2}, t = 1$
Therefore, $I = \int_{\frac{1}{2}}^{1} \frac{d t}{t^{3}} = {∣ ∣ \frac{t^{- 2}}{- 2} ∣ ∣}_{\frac{1}{2}}^{1} = \frac{3}{2}$

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