Solve ∫ wrt x/sqrt{frac{xGMm{x}}}

$$\int \frac { d x } { \sqrt { \frac { x G M m } { x } } }$$

Evaluate

$$\frac{x}{\sqrt{GMm}}+С$$

$$\left(m<0\text{ and }M<0\text{ and }G>0\right)\text{ or }\left(M>0\text{ and }m>0\text{ and }G>0\right)\text{ or }\left(G<0\text{ and }M<0\text{ and }m>0\right)\text{ or }\left(G<0\text{ and }m<0\text{ and }M>0\right)$$

Solution Steps

Cancel out x in both numerator and denominator.

$$\int \frac{1}{\sqrt{GMm}}\mathrm{d}x$$

Find the integral of \frac{1}{\sqrt{GMm}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x}{\sqrt{GMm}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\begin{matrix}\frac{x}{\sqrt{GMm}}+С_{5},&\end{matrix}$$

Differentiate w.r.t. x

$$\frac{1}{\sqrt{GMm}}$$