Answer :

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The solution of the integral of 8 cos(x)+ √3 = 5√3 is  8π√3.

Solution of the given function

The given function is 8 cos(x)+ √3 = 5√3

The function can be simplified as follows;

8 cos(x) + √3 = 5√3

collect similar terms together;

8 cos(x) + √3 - 5√3 = 0

8 cos(x)  - 4√3 = 0

integral of  8 cos(x)  - 4√3  = 8 sin(x) - 4√3x

the value of the integral over (0, 2π)

= 8 sin(0) - 4√3(0) -  (8 sin(2π) - 4√3(2π))

= 0 + 8π√3

= 8π√3

Thus, the solution of the integral of 8 cos(x)+ √3 = 5√3 is  8π√3.

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Answer:

[tex]\text{\textit{x}} \ \ = \ \ \displaystyle\frac{\pi}{6} \ \ \ \text{or} \ \ \ \displaystyle\frac{11\pi}{6}[/tex]

Step-by-step explanation:

Given the trigonometric equation

                               [tex]8 \ \cos{\left(\textit{x}\right)} \ + \ \sqrt{3} \ = \ 5\sqrt{3}[/tex],

hence solving for [tex]\textit{x}[/tex] yields

                                  [tex]8 \ \cos{\left(\textit{x}\right)}\ = \ 5\sqrt{3} \ - \ \sqrt{3} \\ \\ \\ \-\hspace{8px} \cos{\left(x\right)} \ = \ \displaystyle\frac{4\sqrt{3}}{8} \\ \\ \\ \-\hspace{8px} \cos{\left(x\right)} \ = \ \displaystyle\frac{\sqrt{3}}{2} \\ \\ \\ \-\hspace{29px} \textit{x} \ \ \ = \ {\cos}^{-1}\left(\displaystyle\frac{\sqrt{3}}{2}}}\right) \\ \\ \\ \-\hspace{29px} \textit{x} \ \ \ = \ \displaystyle\frac{\pi}{6}[/tex].

Since cosine is positive in both quadrant I and quadrant IV, thus

                                     [tex]\textit{x} \ \ = \ \ 2\pi \ - \ \displaystyle\frac{\pi}{6} \ \ = \ \ \displaystyle\frac{11\pi}{6}[/tex].

Therefore the solutions are

                                               [tex]\textit{x} \ \ = \ \ \displaystyle{\frac{\pi}{6}} \ \ \text{or} \ \ \displaystyle{\frac{11\pi}{6}}[/tex].

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