MCQ
The integral $\int_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}\,x\,\,\cos e{c^{4/3}}\,x\,dx} $ is equal to
- A${3^{5/6}}\, - \,{3^{2/3}}$
- B${3^{5/3}}\, - \,{3^{1/3}}$
- ✓${3^{7/6}}\, - \,{3^{5/6}}$
- D${3^{4/3}}\, - \,{3^{1/3}}$
$=\int \frac{\tan ^{2 / 3} x}{\tan ^{2} x} \cdot \sec ^{2} x \cdot d x$
$=\int \frac{\sec ^{2} x}{\tan ^{4 / 3} x} \cdot d x \quad\left\{\tan x=t, \sec ^{2} x d x=d t\right\}$
$=\int \frac{d t}{\tan ^{4 / 3}}=\frac{t^{-1 / 3}}{-1 / 3}=-3\left(t^{-1 / 3}\right)$
$\Rightarrow 1=-3 \tan (x)^{-1 / 3}$
$\Rightarrow 1=\left.\frac{3}{(\tan x)^{1 / 3}}\right|_{x / 6} ^{\pi / 3}=-3\left(\frac{1}{(\sqrt{3})^{1 / 3}}-(\sqrt{3})^{1 / 3}\right)$
$=3^{7 / 6}-3^{5 / 6}$
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