The indefinite integral I = ∫ ^ 2 xtan x( sec x + tan x) dx ( ^ 5 x...

$I = \dfrac{1}{3}∫ \dfrac{\left(3\sec^{3} xtan x + 3\sec^{2} x\tan^{2} x\right)dx}{\left(\sec^{2} x - \tan^{2} x\right)\left(\sec^{3} x + \tan^{3} x\right)}$
Let $\sec^{3} x + \tan^{3} x = t$
Then, $I = \dfrac{1}{3}∫ \dfrac{dt}{t}$
$= \dfrac{1}{3}\ln \left|t\right| + c$
$= \dfrac{1}{3}\ln \left|\sec^{3} x + \tan^{3} x\right| + c$
$∴ f\left(x\right) = \sec^{3} x + \tan^{3} x$
$⇒ f\left(\dfrac{\pi}{3}\right) = 8 + \sqrt{27}$

$⇒ a = 8, b = 27$
$⇒ b - 3a = 3$