Đáp án:
$1286$ nghiệm
Giải thích các bước giải:
$\quad F(x)= \displaystyle\int f(x)dx$
$\to F(x)= \displaystyle\int\dfrac{1}{\sin^2x\cos^2x}dx$
$\to F(x)=\displaystyle\int\dfrac{4}{\sin^22x}dx$
$\to F(x)=2\displaystyle\int\dfrac{2dx}{\sin^22x}$
$\to F(x)= 2\displaystyle\int\dfrac{d(2x)}{\sin^22x}$
$\to F(x)= -2\cot2x + C$
Ta lại có:
$\quad F\left(\dfrac{\pi}{4}\right) = 1$
$\to -2\cot\dfrac{\pi}{2} + C = 1$
$\to C = 1$
Do đó:
$F(x)= -2\cot2x +1$
Ta có:
$\quad F(x)-1=0$
$\to -2\cot2x = 0$
$\to \cot2x = 0$
$\to 2x = \dfrac{\pi}{2} + k\pi$
$\to x =\dfrac{\pi}{4} + k\dfrac{\pi}{2}\quad (k\in\Bbb Z)$
Mặt khác: $0< x < 2020$
$\to 0 <\dfrac{\pi}{4} + k\dfrac{\pi}{2}< 2020$
$\to -\dfrac12 < k < \dfrac{4040}{\pi} - \dfrac12$
$\to k\in\underbrace{\{0;1;2;\dots;1284;1285\}}_{\text{1286 giá trị k}}$
$\to 1286$ nghiệm
