Solve ∫ (3v^5-v^3/3)dv

$$\int ( 3 v ^ { 5 } - v ^ { 3 / 3 } ) d v$$

Evaluate

$$\frac{v^{6}}{2}-\frac{v^{2}}{2}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 3v^{5}\mathrm{d}v+\int -v\mathrm{d}v$$

Factor out the constant in each of the terms.

$$3\int v^{5}\mathrm{d}v-\int v\mathrm{d}v$$

Since \int v^{k}\mathrm{d}v=\frac{v^{k+1}}{k+1} for k\neq -1, replace \int v^{5}\mathrm{d}v with \frac{v^{6}}{6}. Multiply 3 times \frac{v^{6}}{6}.

$$\frac{v^{6}}{2}-\int v\mathrm{d}v$$

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}. Multiply -1 times \frac{v^{2}}{2}.

$$\frac{v^{6}-v^{2}}{2}$$

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.

$$\frac{v^{6}}{2}-\frac{v^{2}}{2}+С$$

Differentiate w.r.t. v

$$v\left(3v^{4}-1\right)$$