Solve ∫ (from -1 to 3) of 6x wrt x

$$\int_{ -1 }^{ 3 } 6x d x$$

Evaluate

$$24$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6x\mathrm{d}x$$

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

$$6\int x\mathrm{d}x$$

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 6 times \frac{x^{2}}{2}.

$$3x^{2}$$

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.

$$3\times 3^{2}-3\left(-1\right)^{2}$$

Simplify.

$$24$$