Solve ∫ (from - 3 to 24) of 8(4x) wrt x

$$\int _ { - 3 } ^ { 24 } 8 ( 4 x ) d x$$

Evaluate

$$9072$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4\times 8x\mathrm{d}x$$

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

$$4\times 8\int x\mathrm{d}x$$

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}.

$$4\times 8\times \frac{x^{2}}{2}$$

Simplify.

$$16x^{2}$$

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.

$$16\times 24^{2}-16\left(-3\right)^{2}$$

Simplify.

$$9072$$