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Câu hỏi: Họ tất cả các nguyên hàm của hàm số $f(x)=\dfrac{1}{x}\left(1+\dfrac{x}{\cos ^2 x}\right)$ với $x \in(0 ;+\infty) \backslash\left\{\dfrac{\pi}{2}+k \pi, k \in \mathbb{Z}\right\}$ là
A. $\ln x+\tan x+C$.
B. $-\dfrac{1}{x^2}-\tan x+C$.
C. $\ln x-\tan x+C$.
D. $-\dfrac{1}{x^2}+\tan x+C$.
A. $\ln x+\tan x+C$.
B. $-\dfrac{1}{x^2}-\tan x+C$.
C. $\ln x-\tan x+C$.
D. $-\dfrac{1}{x^2}+\tan x+C$.
Ta có:
$
\begin{aligned}
& \text { +) } f(x)=\dfrac{1}{x}\left(1+\dfrac{x}{\cos ^2 x}\right)=\dfrac{1}{x}+\dfrac{1}{\cos ^2 x} . \\
& \text { +) } \int f(x) \mathrm{d} x=\int\left(\dfrac{1}{x}+\dfrac{1}{\cos ^2 x}\right) \mathrm{d} x=\ln |x|+\tan x+C=\ln x+\tan x+C \\
& \text { do } x \in(0 ;+\infty) \backslash\left\{\dfrac{\pi}{2}+k \pi, k \in \mathbb{Z}\right\} .
\end{aligned}
$
$
\begin{aligned}
& \text { +) } f(x)=\dfrac{1}{x}\left(1+\dfrac{x}{\cos ^2 x}\right)=\dfrac{1}{x}+\dfrac{1}{\cos ^2 x} . \\
& \text { +) } \int f(x) \mathrm{d} x=\int\left(\dfrac{1}{x}+\dfrac{1}{\cos ^2 x}\right) \mathrm{d} x=\ln |x|+\tan x+C=\ln x+\tan x+C \\
& \text { do } x \in(0 ;+\infty) \backslash\left\{\dfrac{\pi}{2}+k \pi, k \in \mathbb{Z}\right\} .
\end{aligned}
$
Đáp án A.