Ответ:

cos

2

x=

2

1+cos2x

\begin{gathered}f(x)= \frac{1+cos2x}{2} \\ \\ F(x)= \frac{1}{2}x+ \frac{1}{4}sin2x+C \end{gathered}

f(x)=

2

1+cos2x

F(x)=

2

1

x+

4

1

sin2x+C

Проверка

F`(x)=( \frac{1}{2}x+ \frac{1}{4}sin2x+C)`= \frac{1}{2}+ \frac{1}{4}cos2x\cdot 2+0= \frac{1+cos2x}{2}=F‘(x)=(

2

1

x+

4

1

sin2x+C)‘=

2

1

+

4

1

cos2x⋅2+0=

2

1+cos2x

=

=cos ^{2}x=cos

2

x