I=∫ _π #160; 3 #160; ^ #10;π #160; 6 #160; dx 1+√(tan x #160; ) #160; #10; #160; ......(i) I=∫ _π #160; 3 #160; ^ #1 ...

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Integration of Irrational Algebraic Fractions - 2

∫π/3π/6dx1+√t...

Question

$\int_{π / 3}^{π / 6} \frac{d x}{1 + \sqrt{t a n x}}$

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Solution

$I = \int_{\frac{π}{3}}^{\frac{π}{6}} \frac{d x}{1 + \sqrt{tan x}} . . . . . . (i) I = \int_{\frac{π}{3}}^{\frac{π}{6}} \frac{d x}{1 + \sqrt{tan (\frac{π}{2} - x)}} I = \int_{\frac{π}{3}}^{\frac{π}{6}} \frac{d x}{1 + \sqrt{cot x}} I = \int_{\frac{π}{3}}^{\frac{π}{6}} \frac{\sqrt{tan x} d x}{\sqrt{tan x} + 1} . . . . . . . (i i)$

Adding $(i)$ and $(i i)$

$2 I = \frac{π}{6} - \frac{π}{3} 2 I = - \frac{π}{6} I = - \frac{π}{12}$

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