Question

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

• A. 19
• B. 39
• C. 40
• D. 18

Answer 1

Answer: (A)

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