We are given that
$$
\begin{aligned}
& \int \sqrt{4 \cos ^2(x)-5 \sin ^2(x)} \cos (x) d x \\
& \text { let } I=\int \sqrt{4\left(1-\sin ^2(x)-5 \sin ^2(x)\right.} \cos (x) d x \\
& =\int \sqrt{4-9 \sin ^2(x)} \cdot \cos (x) d x
\end{aligned}
$$
put $3 \sin (x)=t \Rightarrow \cos (x) d x=\frac{1}{3} d t$
$$
\begin{aligned}
& \Rightarrow \int \sqrt{4-\mathrm{t}^2} \frac{1}{3} \mathrm{dt} \\
& \Rightarrow \frac{1}{3}\left[\frac{\mathrm{t}}{2} \sqrt{4-\mathrm{t}^2}+\frac{4}{2} \sin ^{-1}\left(\frac{\mathrm{t}}{2}\right)\right]+\mathrm{c} \\
& \Rightarrow \frac{1}{2} \sin (\mathrm{x}) \sqrt{4-9 \sin ^2(\mathrm{x})}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \cos (\mathrm{x})}{2}\right)+\mathrm{c}
\end{aligned}
$$