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Question
\[\int\limits_0^{\pi/2} \frac{1}{1 + \tan^3 x} dx\]
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Solution
\[Let, I = \int_0^\frac{\pi}{2} \frac{1}{1 + \tan^3 x} d x ..............(1)\]
\[ = \int_0^\frac{\pi}{2} \frac{1}{1 + \tan^3 \left( \frac{\pi}{2} - x \right)} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{1}{1 + co t^3 x} d x ................(2)\]
Adding (1) and (2)
\[2I = \int_0^\frac{\pi}{2} \left[ \frac{1}{1 + \tan^3 x} + \frac{1}{1 + co t^3 x} \right] d x\]
\[ = \int_0^\frac{\pi}{2} \frac{2 + \tan^3 x + co t^3 x}{\left( 1 + \tan^3 x \right)\left( 1 + co t^3 x \right)}dx\]
\[ = \int_0^\frac{\pi}{2} \frac{2 + \tan^3 x + co t^3 x}{2 + \tan^3 x + co t^3 x}dx\]
\[ = \int_0^\frac{\pi}{2} dx \]
\[ = \left( x \right)_0^\frac{\pi}{2} \]
\[ = \frac{\pi}{2}\]
\[Hence, I = \frac{\pi}{4}\]
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