Solve ∫ f(x) wrt x=F(x)

$$\int f ( x ) d x = F ( x )$$

Solve for F

$$\left\{\begin{matrix}F=\frac{fx}{2}+\frac{С}{x}\text{, }&x\neq 0\\F\in \mathrm{R}\text{, }&С=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Fx=\int fx\mathrm{d}x$$

The equation is in standard form.

$$xF=\frac{fx^{2}}{2}+С$$

Divide both sides by x.

$$\frac{xF}{x}=\frac{\frac{fx^{2}}{2}+С}{x}$$

Dividing by x undoes the multiplication by x.

$$F=\frac{\frac{fx^{2}}{2}+С}{x}$$

Divide \frac{fx^{2}}{2}+С by x.

$$F=\frac{fx}{2}+\frac{С}{x}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{F}{x}-\frac{С}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&С=0\text{ and }x=0\end{matrix}\right.$$