Solve ∫ (from 0 to x) of 3cdc

$$\int _ { 0 } ^ { x } 3 c d c$$

Evaluate

$$\frac{3x^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3c\mathrm{d}c$$

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

$$3\int c\mathrm{d}c$$

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

$$\frac{3c^{2}}{2}$$

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.

$$\frac{3}{2}x^{2}-\frac{3}{2}\times 0^{2}$$

Simplify.

$$\frac{3x^{2}}{2}$$

Differentiate w.r.t. x

$$3x$$