Solve ∫ (1/4V^4+V^2+V)dv

$$\int ( \frac { 1 } { 4 } V ^ { 4 } + V ^ { 2 } + V ) d v$$

Evaluate

$$\frac{Vv\left(V^{3}+4V+4\right)}{4}+С$$

Solution Steps

Find the integral of \frac{V^{4}}{4}+V^{2}+V using the table of common integrals rule \int a\mathrm{d}v=av.

$$\left(\frac{V^{4}}{4}+V^{2}+V\right)v$$

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.

$$\left(\frac{V^{4}}{4}+V^{2}+V\right)v+С$$

Differentiate w.r.t. V

$$v\left(V^{3}+2V+1\right)$$