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Q)

Evaluate the following problems using properties of integration: $\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{dx}{1+\sqrt{\tan x}}$

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A)
Toolbox:
  • $\int \limits_a^b f(x)dx=\int \limits_a^b f(a+b-x)dx$
$\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{dx}{1+\sqrt{\tan x}}$
Step 1:
$I=\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{dx}{1+\sqrt{\tan x}}$
Step 2:
$\qquad=\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{dx}{1+\sqrt{\tan (\Large\frac{\pi}{6}+\frac{\pi}{3}-x)}}$
$\qquad=\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{dx}{1+\sqrt {\cot x}}$
Step 3:
$2I=\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{dx}{1+\sqrt{\Large\frac{\sin x}{\cos x}}}$$+\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{\sqrt {\sin x}dx}{\sqrt {\sin x}+\sqrt {\cos x}}$
$\qquad=\int\limits_{\large\frac{\pi}{6}}^{\large\frac{\pi}{3}}\large\frac{(\sqrt {\cos x}+\sqrt { \sin x})}{(\sqrt{\cos x}+\sqrt {\sin x})}$$dx$
$\qquad= x \bigg]_{\large\frac{\pi}{6}} ^{\large\frac{\pi}{3}}$
$\qquad= \large\frac{\pi}{3} -\large\frac{\pi}{6}=\frac{\pi}{6}$
Step 4:
=>$I=\large\frac{\pi}{12}$
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