Solve ∫ (from 2 to C) of v^3dv

$$\int _ { 2 } ^ { C } v ^ { 3 } d v$$

Evaluate

$$\frac{C^{4}}{4}-4$$

Solution Steps

Evaluate the indefinite integral first.

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

Since \int v^{k}\mathrm{d}v=\frac{v^{k+1}}{k+1} for k\neq -1, replace \int v^{3}\mathrm{d}v with \frac{v^{4}}{4}.

$$\frac{v^{4}}{4}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{C^{4}}{4}-\frac{2^{4}}{4}$$

Simplify.

$$\frac{C^{4}}{4}-4$$

Differentiate w.r.t. C

$$C^{3}$$