Họ nguyên hàm của hàm số \(f\left( x \right) = \frac{1}{x} + \sin x\) là

A. \[{x^2} + C\].

B. \[{x^2} + 6x + C\].

C. \[2{x^2} + C\].

D. \[2{x^2} + 6x + C\].

A. \[\frac{1}{5}{x^5} + \frac{1}{3}{x^3} + C\]

B. \[{x^4} + {x^2} + C\]

C. \[{x^5} + {x^3} + C\].

D. \[4{x^3} + 2x + C\]

A. \[\frac{1}{{12}}{x^4} - \frac{2}{3}{x^3} + \frac{{{x^2}}}{2} + C\].

B. \(\frac{1}{9}{x^4} - \frac{2}{3}{x^3} + \frac{{{x^2}}}{2} - 2024x + C\).

C. \(\frac{1}{{12}}{x^4} - \frac{2}{3}{x^3} + \frac{{{x^2}}}{2} - 2024x + C\).

D. \(\frac{1}{9}{x^4} + \frac{2}{3}{x^3} - \frac{{{x^2}}}{2} - 2024x + C\).

A. \[F\left( x \right)\, = \,\frac{{{x^4}}}{4}\, - \,6{x^3}\, + \,\frac{{11}}{2}{x^2} - \,6x\, + \,C\].B. \[F\left( x \right)\, = \,{x^4}\, + \,6{x^3}\, + \,11{x^2} + \,6x\, + \,C\].

C. \[F\left( x \right)\, = \,\frac{{{x^4}}}{4}\, + 2{x^3}\, + \,\frac{{11}}{2}{x^2} + \,6x\, + \,C\].D. \[F\left( x \right)\, = \,{x^3}\, + \,6{x^2}\, + \,11{x^2} + \,6x\, + \,C\].

B. A. \(F\left( x \right) = 2\cos \frac{x}{2} + C\)

C. B. \(F\left( x \right) = \frac{1}{2}\left( {1 + \sin x} \right) + C\)

D. C. \(F\left( x \right) = 2\sin \frac{x}{2} + C\)

A. D. \(F\left( x \right) = \frac{1}{2}\left( {1 - \sin x} \right) + C\)

A. \(\frac{{{x^2}}}{2} + \sin x + C\).

B. \(\frac{{{x^2}}}{2} - \cos x + C\).

C. \(\frac{{{x^2}}}{2} - \sin x + C\).

D. \(\frac{{{x^2}}}{2} + \cos x + C\).

A. \(f\left( x \right) = - \sin x\).

B. \(f\left( x \right) = - \cos x\).

C. \(f\left( x \right) = \sin x\).

D. \(f\left( x \right) = \cos x\).