Solve ∫ (from 0 to 4) of 12x wrt x

$$\int _ { 0 } ^ { 4 } 12 x d x$$

Evaluate

$$96$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 12x\mathrm{d}x$$

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

$$12\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 12 times \frac{x^{2}}{2}.

$$6x^{2}$$

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.

$$6\times 4^{2}-6\times 0^{2}$$

Simplify.

$$96$$