The value of / 6 / 6 ( + 4 x 11 1 - sin | x | + / 6 ) ? d x is equal to: [2026]

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Solution

Q.
The value of $\int_{- π / 6}^{π / 6} (\frac{π + 4 x^{11}}{1 - \sin (| x | + π / 6)}) d x$ is equal to: [2026]

1 $8 π$

2 $2 π$

3 $6 π$

4 $4 π$

Ans.
(4)

$= 2 π \int_{0}^{π / 6} \frac{1}{1 - \sin (x + \frac{π}{6})} d x let x + \frac{π}{6} = t \Rightarrow d x = d t$

$= 2 π \int_{π / 6}^{π / 3} \frac{d t}{1 - \sin t} = 2 π \int_{π / 6}^{π / 3} \frac{1 + \sin t}{\cos^{2} t} d t$

$= 2 π [\int_{π / 6}^{π / 3} \sec^{2} t d t + \int_{π / 6}^{π / 3} \sec t \tan t d t]$

$= 2 π [{(\tan t)}_{π / 6}^{π / 3} + {(\sec t)}_{π / 6}^{π / 3}]$

$= 2 π [(\sqrt{3} - \frac{1}{\sqrt{3}}) + (2 - \frac{2}{\sqrt{3}})]$

$= 2 π [\sqrt{3} + 2 - \sqrt{3}] = 4 π$

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