## Subject IV. (20 points)

Determine the function $f: \mathbb{R} \rightarrow \mathbb{R}_{+}^{*}$ such that the function $g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=f(x)(\sin 2 x+4 \cos x)$

admits the primitive $G: \mathbb{R} \rightarrow \mathbb{R}, G(x)=\frac{f(x)}{2+\sin x}$.

Prof. Gheorghe Lobont, National College "Mihai Viteazul" Turda

All subjects are mandatory. 10 points are awarded by default.

SUCCESS!

Effective working time - 3 hours.

## Grading Scale for Grade 12 (OLM 2014 - local stage)

## Of. $10 \text{p}$
## Subject IV.

The relationship $G^{\prime}(x)=g(x)$ leads to $\frac{f^{\prime}(x)(2+\sin x)-f(x) \cos x}{(2+\sin x)^{2}}=f(x)(\sin 2 x+4 \cos x)$ and from here we obtain

$\frac{f^{\prime}(x)}{f(x)}=\frac{\cos x}{2+\sin x}+2 \cos x(2+\sin x)^{2}$.

(10p)

It follows that $\int \frac{f^{\prime}(x)}{f(x)} d x=\int \frac{\cos x}{2+\sin x} d x+2 \int \cos x(2+\sin x)^{2} d x$ and therefore we obtain

$\ln f(x)=\ln (2+\sin x)+\frac{2}{3}(2+\sin x)^{3}+\ln |\mathfrak{C}| \Leftrightarrow f(x)=|\mathfrak{C}| \cdot e^{\frac{2}{3}(2+\sin x)^{3}} \cdot(2+\sin x)$.