Question

The value of the integral $\int_{\pi/6}^{\pi/3}\frac{dx}{1 + \tan^{5}x}$ is

• A. 1
• B. π/12
• C. π/6
• D. None of these

Answer 1

Answer: (B)

π/12

Answer 2

Using the property $\int_{a}^{b}{}$f (x) d x = $\int_{a}^{b}{}$f (a + b – x) dx, the given integral

I = $\int_{\pi/6}^{\pi/3}\frac{dx}{1 + \tan^{5}x}$ = $\int_{\pi/6}^{\pi/3}\frac{dx}{1 + \tan^{5}\left( \frac{\pi}{3} + \frac{\pi}{6} - x \right)}$= $\int_{\pi/6}^{\pi/3}\frac{dx}{1 + \cot^{5}x}$.

Hence 2 I = $\int_{\pi/6}^{\pi/3}{dx}$⇒ I = $\frac { 1 } { 2 }$ $\left( \frac{\pi}{3} - \frac{\pi}{6} \right)$ = $\frac{\pi}{12}$