Solve ∫ mvdv

$$\int m v d v$$

Evaluate

$$\frac{mv^{2}}{2}+С$$

Solution Steps

Factor out the constant using \int af\left(v\right)\mathrm{d}v=a\int f\left(v\right)\mathrm{d}v.

$$m\int v\mathrm{d}v$$

Since \int v^{k}\mathrm{d}v=\frac{v^{k+1}}{k+1} for k\neq -1, replace \int v\mathrm{d}v with \frac{v^{2}}{2}.

$$m\times \frac{v^{2}}{2}$$

Simplify.

$$\frac{mv^{2}}{2}$$

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

$$\frac{mv^{2}}{2}+С$$

Differentiate w.r.t. m

$$\frac{v^{2}}{2}$$