Question

Evaluate the integral

${\int }_{\pi /6}^{\pi /3}\frac{dx}{1+{\tan }^{11}x}$

• A. $\frac{\pi }{12}$
• B. $\frac{\pi }{3}$
• C. $1$
• D. None of the above

Answer 1

Answer: (A)

$\frac{\pi }{12}$

$I={\int }_{\pi /6}^{\pi /3}\frac{dx}{1+{\tan }^{11}x}$

$={\int }_{\pi /6}^{\pi /3}\frac{dx}{1+{\tan }^{11}(\frac{\pi }{6}+\frac{\pi }{3}-x)}$

$={\int }_{\pi /6}^{\pi /3}\frac{dx}{1+{\cot }^{11}x}$

$={\int }_{\pi /6}^{\pi /3}\frac{{\tan }^{11}xdx}{1+{\tan }^{11}x}={\int }_{\pi /6}^{\pi /3}dx-{\int }_{\pi /6}^{\pi /3}\frac{dx}{1+{\tan }^{11}x}$

$\Rightarrow I=\frac{\pi }{6}-I$

$\Rightarrow I=\frac{\pi }{12}$