Question

Given that $\int \frac{\sin ^3 x-x \sin ^2 x \cdot \cos x}{\cos ^3 x} d x=f(x)+C$, where $C$ is constant of integration. If $f(\pi)=\frac{\pi^2}{2}$, then the value of $[f(2 \pi)]$ equals.
[Note: Where $[ k ]$ denotes greatest integer function less than or equal to $k$ ]

• A. 19
• B. 39
• C. 40
• D. 18

Answer 1

Answer: (A)

$\int\left(\tan ^3 x-x \tan ^2 x\right) d x=\int \tan ^2 x(\tan x-x) d x$
$\text { Let }\tan x-x=t $
$\left(\sec ^2 x-1\right) d x=d t$
$ \int tdt =\frac{ t ^2}{2}+ C =\frac{(\tan x - x )^2}{2}+ C $
$ f ( x )=\frac{(\tan x - x )^2}{2} $
$ ( f (2 \pi))=2 \pi^2 \approx 19.73 $
$\therefore [ f (2 \pi)]=19$