7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$=\int \frac{\tan ^{2 / 3} x}{\tan ^{2} x} \cdot \sec ^{2} x \cdot d x$

$=\int \frac{\sec ^{2} x}{\tan ^{4 / 3} x} \cdot d x \quad\left\{\tan x=t, \sec ^{2} x d x=d t\right\}$

$=\int \frac{d t}{\tan ^{4 / 3}}=\frac{t^{-1 / 3}}{-1 / 3}=-3\left(t^{-1 / 3}\right)$

$\Rightarrow 1=-3 \tan (x)^{-1 / 3}$

$\Rightarrow 1=\left.\frac{3}{(\tan x)^{1 / 3}}\right|_{x / 6} ^{\pi / 3}=-3\left(\frac{1}{(\sqrt{3})^{1 / 3}}-(\sqrt{3})^{1 / 3}\right)$

$=3^{7 / 6}-3^{5 / 6}$