Question

\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^3 x} dx\]  is equal to

Options

• 0

• 1

• π/2

• π/4

Answer 1

 π/4

 

We have
\[ I = \int_0^\frac{\pi}{2} \frac{1}{1 + \cot^3 x} d x . . . . . \left( 1 \right)\]
\[ = \int_0^\frac{\pi}{2} \frac{1}{1 + \cot^3 \left( \frac{\pi}{2} - x \right)} d x \]
\[ \therefore I = \int_0^\frac{\pi}{2} \frac{1}{1 + \tan^3 x} d x . . . . . \left( 2 \right)\]
\[\text{Adding} \left( 1 \right) and \left( 2 \right) \text{we get}\]
\[2I = \int_0^\frac{\pi}{2} \left[ \frac{1}{1 + co t^3 x} + \frac{1}{1 + \tan^3 x} \right] d x\]

\[= \int_0^\frac{\pi}{2} \left[ \frac{1 + \tan^3 x + 1 + co t^3 x}{\left( 1 + co t^3 x \right)\left( 1 + \tan^3 x \right)} \right] dx\]
\[ = \int_0^\frac{\pi}{2} \left[ \frac{2 + \tan^3 x + co t^3 x}{1 + \tan^3 x + co t^3 x + co t^3 x \tan^3 x} \right]dx\]
\[ = \int_0^\frac{\pi}{2} \left[ \frac{2 + \tan^3 x + co t^3 x}{1 + \tan^3 x + co t^3 x + 1} \right]dx\]
\[ = \int_0^\frac{\pi}{2} \left[ \frac{2 + \tan^3 x + co t^3 x}{2 + \tan^3 x + co t^3 x} \right] dx\]
\[ = \int_0^\frac{\pi}{2} [1]dx\]
\[ = \left[ x \right]_0^\frac{\pi}{2} \]
\[ = \frac{\pi}{2}\]
\[Hence\ I = \frac{\pi}{4}\]