Solve ∫ (from - 2 to 3) of 4(2x) wrt x

$$\int _ { - 2 } ^ { 3 } 4 ( 2 x ) d x$$

Evaluate

$$20$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2\times 4x\mathrm{d}x$$

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

$$2\times 4\int x\mathrm{d}x$$

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}.

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

Simplify.

$$4x^{2}$$

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.

$$4\times 3^{2}-4\left(-2\right)^{2}$$

Simplify.

$$20$$