Đáp án:

Giải thích các bước giải:

 $f(x)=(2\ \text{tan}\ x-\text{cot}\ x)^2=4\ \text{tan}^2\ x+\text{cot}^2\ x-4$

$⇒f(x)=4\left(\dfrac{1}{\text{cos}^2\ x}-1\right)+\left(\dfrac{1}{\text{sin}^2\ x}-1\right)-4$

$⇒f(x)=\dfrac{4}{\text{cos}^2\ x}+\dfrac{1}{\text{sin}^2\ x}-9$

$⇒F(x)=4\displaystyle\int\frac{dx}{\cos^2 x}+\int\frac{dx}{\sin^2 x}-9x+C
=4\tan x-\cot x-9x+C$

Điều kiện $F\left(\dfrac{\pi}{4}\right)=3-\dfrac{9\pi}{4},\ \text{tan}\ \dfrac{\pi}{4}=\text{cot}\ \dfrac{\pi}{4}=1$

$⇒4-1-\dfrac{9\pi}{4}+C=3-\dfrac{9\pi}{4}$

$⇒C=0$

$⇒F(x)=4\ \text{tan}\ x-\text{cot}\ x-9x$

$⇒abc=36$