Solve ∫ (from 2 to 4) of x+4x wrt x

$$\int_{ 2 }^{ 4 } x+4x d x$$

Evaluate

$$30$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x+4x\mathrm{d}x$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int 4x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x\mathrm{d}x+4\int x\mathrm{d}x$$

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}.

$$\frac{x^{2}}{2}+4\int x\mathrm{d}x$$

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}. Multiply 4 times \frac{x^{2}}{2}.

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

Simplify.

$$\frac{5x^{2}}{2}$$

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.

$$\frac{5}{2}\times 4^{2}-\frac{5}{2}\times 2^{2}$$

Simplify.

$$30$$